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IB Mathematics AI SL - Questionbank

Sequences & Series

Arithmetic & Geometric Sequences & Series, Finding Terms & Sum of Terms, Sigma Notation, Applications...

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Question 1

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easy

[Maximum mark: 6]

The 1515th term of an arithmetic sequence is 2121 and the common difference is 4-4.

  1. Find the first term of the sequence. [2]

  2. Find the 2929th term of the sequence. [2]

  3. Find the sum of the first 4040 terms of the sequence. [2]

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Question 2

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easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

an=1,5,10,15,cn=1.5,3,4.5,6,bn=12,23,34,45,dn=2,1,12,14,\begin{array}{rcccccl} a_n &=& 1,\,5,\,10,\,15,\,\dots &\,\hspace{4em}\,& c_n &=& 1.5,\,3,\,4.5,\,6,\,\dots \\[12pt] b_n &=& \dfrac{1}{2},\,\dfrac{2}{3},\,\dfrac{3}{4},\,\dfrac{4}{5},\,\dots &\,\hspace{4em}\,& d_n &=& 2,\,1,\,\dfrac{1}{2},\,\dfrac{1}{4},\,\dots \end{array}
  1. State which sequence is arithmetic and find the common difference of the sequence. [2]

  2. State which sequence is geometric and find the common ratio of the sequence.[2]

  3. For the geometric sequence find the exact value of the eighth term. Give your answer as a fraction. [2]

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Question 3

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[Maximum mark: 6]

An arithmetic sequence has u1=40u_1 = 40, u2=32u_2 = 32, u3=24u_3 = 24.

  1. Find the common difference, dd. [2]

  2. Find u8u_8. [2]

  3. Find S8S_8. [2]

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Question 4

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easy

[Maximum mark: 6]

Only one of the following four sequences is arithmetic and only one of them is geometric.

an=13,14,15,16,cn=3,1,13,19,bn=2.5,5,7.5,10,dn=1,3,6,10,\begin{array}{rcccccl} a_n &=& \dfrac{1}{3},\,\dfrac{1}{4},\,\dfrac{1}{5},\,\dfrac{1}{6},\,\dots &\,\hspace{4em}\,& c_n &=& 3,\,1,\,\dfrac{1}{3},\,\dfrac{1}{9},\,\dots \\[12pt] b_n &=& 2.5,\,5,\,7.5,\,10,\,\dots &\,\hspace{4em}\,& d_n &=& 1,\,3,\,6,\,10,\,\dots \end{array}
  1. State which sequence is arithmetic and find the common difference of the sequence. [2]

  2. State which sequence is geometric and find the common ratio of the sequence.[2]

  3. For the geometric sequence find the exact value of the sixth term. Give your answer as a fraction. [2]

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Question 5

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[Maximum mark: 6]

An arithmetic sequence has u1=12u_1 = 12, u2=21u_2 = 21, u3=30u_3 = 30.

  1. Find the common difference, dd. [2]

  2. Find u10u_{10}. [2]

  3. Find S10S_{10}. [2]

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Question 6

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easy

[Maximum mark: 7]

Brendan is training for a long distance bike race.

In week 11 of his training he cycled 2222\,km. In week 22 he cycled 3434\,km. This pattern continues, with him cycling an extra 1212\,km per week.

The distances Brendan cycled in the first 55 weeks of training is shown in the following table.

Screenshot 2023-08-31 at 2.15.24 PM

  1. Calculate how far he cycles in the 1717th week of his training. [3]

  2. In total how far has he cycled after 1717 weeks? [2]

  3. Find the mean distance per week he cycled over 17 weeks. [2]

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Question 7

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easy

[Maximum mark: 6]

A geometric sequence has u1=5u_1 =5, u2=1u_2 = -1 and u3=15u_3 = \dfrac{1}{5}.

  1. Find the common ratio, rr. [2]

  2. Find the exact value of u7u_{7}. [2]

  3. Find the exact value of S7S_{7}. [2]

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Question 8

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[Maximum mark: 6]

Emily starts reading Leo Tolstoy's War and Peace on the 11st of February. The number of pages she reads each day increases by the same number on each successive day.

c94a768fb53af8987d3e1115bdd47ee0b1976776.svg

  1. Calculate the number of pages Emily reads on the 1414th of February. [3]

  2. Find the exact total number of pages Emily reads in the 2828 days of February.[3]

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Question 9

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easy

[Maximum mark: 6]

A geometric sequence has 2020 terms, with the first four terms given below.

418.5,279,186,124,\begin{aligned} 418.5,\hspace{0.25em} 279,\hspace{0.25em} 186,\hspace{0.25em} 124,\hspace{0.25em}\dots \\ \end{aligned}
  1. Find rr, the common ratio of the sequence. Give your answer as a fraction. [1]

  2. Find u5u_5, the fifth term of the sequence. Give your answer as a fraction. [1]

  3. Find the smallest term in the sequence that is an integer. [2]

  4. Find S10S_{10}, the sum of the first 1010 terms of the sequence. Give your answer correct to one decimal place. [2]

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Question 10

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easy

[Maximum mark: 6]

A tennis ball bounces on the ground nn times. The heights of the bounces, h1,h2,h3,,hn,h_1, h_2, h_3, \dots,h_n, form a geometric sequence. The height that the ball bounces the first time, h1h_1, is 8080 cm, and the second time, h2h_2, is 6060 cm.

  1. Find the value of the common ratio for the sequence. [2]

  2. Find the height that the ball bounces the tenth time, h10h_{10}. [2]

  3. Find the total distance travelled by the ball during the first six bounces (up and down). Give your answer correct to 22 decimal places. [2]

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Question 11

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easy

[Maximum mark: 6]

The table shows the first four terms of three sequences: unu_n, vnv_n, and wnw_n.

c39694c1cf7513ffce115791e6b0f1c54c230963.svg

  1. State which sequence is

    1. arithmetic;

    2. geometric. [2]

  2. Find the sum of the first 5050 terms of the arithmetic sequence. [2]

  3. Find the exact value of the 1313th term of the geometric sequence. [2]

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Question 12

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easy

[Maximum mark: 6]

The table below shows the first four terms of three sequences: unu_n, vnv_n, and wnw_n.

6896afb03e54861ed9a71ba4f129a85ea32667d8.svg

  1. State which sequence is

    1. arithmetic;

    2. geometric. [2]

  2. Find the exact value of the sum of the first 3535 terms of the arithmetic
    sequence. [2]

  3. Find the exact value of the 1010th term of the geometric sequence. [2]

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Question 13

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[Maximum mark: 6]

In this question give all answers correct to the nearest whole number.

A population of goats on an island starts at 232232. The population is expected
to increase by 1515 % each year.

  1. Find the expected population size after:

    1. 1010 years;

    2. 2020 years. [4]

  2. Find the number of years it will take for the population to reach 1500015\hspace{0.15em}000. [2]

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Question 14

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[Maximum mark: 6]

On the first day of September, 20192019, Gloria planted 55 flowers in her garden. The number of flowers, which she plants at every day of the month, forms an arithmetic sequence. The number of flowers she is going to plant in the last day of September is 6363.

  1. Find the common difference of the sequence. [2]

  2. Find the total number of flowers Gloria is going to plant during September.[2]

  3. Gloria estimated she would plant 10001000 flowers in the month of September. Calculate the percentage error in Gloria's estimate. [2]

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Question 15

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easy

[Maximum mark: 6]

The fifth term, u5u_5, of a geometric sequence is 375375. The sixth term, u6u_6, of the sequence is 7575.

  1. Write down the common ratio of the sequence. [1]

  2. Find u1u_1. [2]

The sum of the first kk terms in the sequence is 292968292\hspace{0.15em}968.

  1. Find the value of kk. [3]

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Question 16

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[Maximum mark: 6]

The third term, u3u_3, of an arithmetic sequence is 77. The common difference of
the sequence, dd, is 33.

  1. Find u1u_1, the first term of the sequence. [2]

  2. Find u60u_{60}, the 6060th term of sequence. [2]

The first and fourth terms of this arithmetic sequence are the first two terms
of a geometric sequence.

  1. Calculate the sixth term of the geometric sequence. [2]

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Question 17

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[Maximum mark: 6]

A 33D printer builds a set of 4949 Eif\text{f}fel Tower Replicas in different sizes. The height of the largest tower in this set is 6464 cm. The heights of successive smaller towers are 9595 % of the preceding larger tower, as shown in the diagram below.

AI110

  1. Find the height of the smallest tower in this set. [3]

  2. Find the total height if all 4949 towers were placed one on top of another. [3]

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Question 18

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[Maximum mark: 6]

The fourth term, u4u_4, of a geometric sequence is 135135. The fifth term, u5u_5, is 8181.

  1. Find the common ratio of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Calculate the sum of the first 2020 terms of the sequence. [2]

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Question 19

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[Maximum mark: 6]

The fifth term, u5u_5, of a geometric sequence is 125125. The sixth term, u6u_6, is 156.25156.25.

  1. Find the common ratio of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Calculate the sum of the first 1212 terms of the sequence. [2]

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Question 20

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[Maximum mark: 6]

The fifth term, u5u_5, of an arithmetic sequence is 55. The eighth term, u8u_8, of the same sequence is 1414.

  1. Find dd, the common difference of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Find S100S_{100}, the sum of the first 100100 terms of the sequence. [2]

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Question 21

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[Maximum mark: 6]

The fifth term, u5u_5, of an arithmetic sequence is 2525. The eleventh term, u11u_{11}, of the same sequence is 4949.

  1. Find dd, the common difference of the sequence. [2]

  2. Find u1u_1, the first term of the sequence. [2]

  3. Find S100S_{100}, the sum of the first 100100 terms of the sequence. [2]

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Question 22

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[Maximum mark: 15]

Charles has a New Years Resolution that he wants to be able to complete 100100 pushups in one go without a break. He sets out a training regime whereby he completes 2020 pushups on the first day, then adds 55 pushups each day thereafter.

  1. Write down the number of pushups Charles completes
    1. on the 44th training day;

    2. on the nnth training day. [3]

On the kkth training day Charles completes 100100 pushups for the first time.

  1. Find the value of kk. [2]

  2. Calculate the total number of pushups Charles completes on the first 1010 training days. [4]

Charles is also working on improving his long distance swimming in preparation for an Iron Man event in 1212 weeks time. He swims a total of 1000010\hspace{0.15em}000 metres in the first week, and plans to increase this by 1010 % each week up until the event.

  1. Find the distance Charles swims in the 66th week of training. [3]

  2. Calculate the total distance Charles swims until the event. [3]

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Question 23

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[Maximum mark: 6]

The second and the third terms of a geometric sequence are u2=3u_2 = 3 and u3=6u_3 = 6.

  1. Find the value of rr, the common ratio of the sequence. [2]

  2. Find the value of u6u_6. [2]

  3. Find the largest value of nn for which unu_n is less than 10410^4.[2]

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Question 24

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[Maximum mark: 12]

The sum of the first nn terms of an arithmetic sequence, Sn=u1+u2+u3++unS_n = u_1 + u_2 + u_3 + \dots + u_n, is given by Sn=2n2+nS_n = 2n^2 + n.

  1. Write down the values of S1S_1 and S2S_2. [2]

  2. Write down the values of u1u_1 and u2u_2. [2]

  3. Find dd, the common difference of the sequence. [1]

  4. Find u10u_{10}, the tenth term of the sequence. [2]

  5. Find the greatest value of nn, for which unu_n is less than 100100. [3]

  6. Find the value of nn, for which SnS_n is equal to 12751275. [2]

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Question 25

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[Maximum mark: 6]

The Australian Koala Foundation estimates that there are about 4500045\hspace{0.15em}000 koalas left in the wild in 20192019. A year before, in 20182018, the population of koalas was estimated as 5000050\hspace{0.15em}000. Assuming the population of koalas continues to decrease by the same percentage each year, find:

  1. the exact population of koalas in 20222022; [3]

  2. the number of years it will take for the koala population to reduce to half of its number in 20182018. [3]

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Question 26

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[Maximum mark: 6]

A battalion is arranged, per row, according to an arithmetic sequence. There are 2424 troops in the third row and 4242 troops in the sixth row.

  1. Find the first term and the common difference of this arithmetic sequence. [3]

There are 1515 rows in the battalion.

  1. Find the total number of troops in the battalion. [3]

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Question 27

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[Maximum mark: 6]

Melinda has $300000\$300\hspace{0.15em}000 in a private foundation. Each year she donates 10%10\hspace{0.05em}\% of the money remaining in her private foundation to charity.

  1. Find the maximum number of years Melinda can donate to charity while keeping at least $100000\$100\hspace{0.15em}000 in the private foundation. [3]

Bill invests $400000\$400\hspace{0.15em}000 in a bank account that pays a nominal interest rate of 44 %, compounded quarterly, for ten years.

  1. Calculate the value of Bill's investment at the end of this time. Give your answer correct to the nearest dollar. [3]

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Question 28

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[Maximum mark: 15]

Towards the end of 2004, a theatre company upgraded their auditorium and installed new comfortable ergonomic chairs for the audience.

After the redesign, there were 2020 seats in the first row and each subsequent row had three more seats than the previous row.

  1. If the auditorium had a total of 1616 rows, find
    1. the total number of seats in the last row.

    2. the total number of seats in the auditorium. [5]

The auditorium reopened for performances at the start of 2005. The average number of visitors per show during that year was 500500. In 2006, the average number of visitors per show increased by 5%5\%.

  1. Find the average number of visitors per show in 2006. [1]

The average number of visitors per show continued to increase by 5%5\% each year.

  1. Determine the first year in which the total number of visitors to a show exceeded the seating capacity of the auditorium. [5]

The theatre company hosts 2525 shows per year.

  1. Determine the total number of visitors that attended the auditorium from when it opened in 2005 until the end of 2011. Round your answer correct to the nearest integer. [4]

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Question 29

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[Maximum mark: 6]

The first term of an arithmetic sequence is 2424 and the common difference is 1616.

  1. Find the value of the 6262 nd term of the sequence. [2]

The first term of a geometric sequence is 88. The 44th term of the geometric sequence is equal to the 1313th term of the arithmetic sequence given above.

  1. Write down an equation using this information. [2]

  2. Calculate the common ratio of the geometric sequence. [2]

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Question 30

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[Maximum mark: 15]

A ball is dropped from the top of the Eiffel Tower, 324324 metres from the ground. The ball falls a distance of 4.94.9 metres during the first second, 14.714.7 metres during the next second, 24.524.5 metres during the third second, and so on. The distances that the ball falls each second form an arithmetic sequence.

    1. Find dd, the common difference of the sequence.

    2. Find u5u_5, the fifth term of the sequence. [2]

  1. Find S6S_6, the sum of the first 66 terms of the sequence. [2]

  2. Find the time the ball will take to reach the ground. Give your answer in seconds correct to one decimal place. [3]

Assuming the ball is dropped another time from a much higher height than of the Eiffel Tower,

  1. find the distance the ball travels from the start of the 1010th second to the end of the 1515th second. [3]

The Eiffel Tower in Paris, France was opened in 18891889, and 1.91.9 million visitors ascended it during that first year. The number of people who visited the tower the following year (18901890) was 22 million.

  1. Calculate the percentage increase in the number of visitors from 18891889 to 18901890. Give your answer correct to one decimal place. [2]

  2. Use your answer to part (e) to estimate the number of visitors in 19001900, assuming that the number of visitors continues to increase at the same percentage rate. [3]

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Question 31

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[Maximum mark: 13]

On September 1st, an orchard commences the process of harvesting 3636 hectares of apple trees. At the end of September 4th, there were 3030 hectares remaining to be harvested, and at the end of September 8th, there were 2424 hectares remaining. Assuming that the number of hectares harvested each day is constant, the total number of hectares remaining to be harvested can be described by an arithmetic sequence.

  1. Find the number of hectares of apple trees that are harvested each day. [3]

  2. Determine the number of hectares remaining to be harvested at the end of September 1st. [1]

  3. Determine the date on which the harvest will be complete. [2]

In 2021 the orchard sold their apple crop for $220000\$220\,000. It is expected that the selling price will then increase by 3.2%3.2\% annually for the next 77 years.

  1. Determine the amount of money the orchard will earn for their crop in 2026. Round your answer to the nearest dollar. [3]

    1. Find the value of n=18(220000×1.032n1)\displaystyle\sum_{n=1}^8 \big(220\hspace{0.15em}000 \times 1.032^{n-1}\big). Round your answer to the nearest integer.

    2. Describe, in context, what the value in part (e) (i) represents. [3]

  2. Comment on whether it is appropriate to model this situation in terms of a geometric sequence. [1]

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Question 32

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[Maximum mark: 16]

The number of seats in an auditorium follows a regular pattern where the first row has u1u_1 seats, and the amount increases by the same amount, dd, each row. In the fifth row, there are 6262 seats and in the thirteenth row there are 8686 seats.

  1. Write down an equation, in terms of u1u_1 and dd, for the amount of seats

    1. in the fifth row.

    2. in the thirteenth row.[2]

  2. Find the value of u1u_1 and dd.[2]

  3. Calculate the total number of seats if the auditorium has 20 rows.[3]

The cost of the ticket for a musical held at the auditorium is inversely proportional to the seat's row. The price for a seat in the first row is $120 dollars, and the price decreases 3%3\% each row. Thus, the value of the ticket for seats in the second row is $116.40 and $112.91 in the third one, etc.

    1. Find the price of the ticket for a seat in the fifth row, rounding your answer to two decimal places.

    2. Find the row of the seat at which the price of a ticket first falls below $70.

    3. Find the total revenue the auditorium generates by tickets sales if 40 seats in each of the 20 rows are sold. Give your answer rounded to the nearest dollar.[9]

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Question 33

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[Maximum mark: 7]

Two college students, David and Lisa, decide to invest money they have saved from working part-time jobs. David's investment strategy results in an increase of his investment amount by $1000\$ 1\,000 each year. Lisa's investment strategy results in her investment amount increasing by 5%5 \% each year.

At the start of the second year of investing, David's total investment amount is $21000\$21\,000 and Lisa's is $11655\$11\,655.

  1. Calculate
    1. the original amount David invested.

    2. the original amount Lisa invested.[4]

During a certain year, nn, Lisa's investment amount becomes larger than David's amount for the first time.

  1. Find the value of nn. [3]

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Question 34

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[Maximum mark: 14]

Georgia is on vacation in Costa Rica. She is in a hot air balloon over a lush jungle in Muelle.

When she leans forward to see the treetops, she accidentally drops her purse. The purse falls down a distance of 44 metres during the first second, 1212 metres during the next second, 2020 metres during the third second and continues in this way. The distances that the purse falls during each second forms an arithmetic sequence.

    1. Write down the common difference, dd, of this arithmetic sequence.

    2. Write down the distance the purse falls during the fourth second. [2]

  1. Calculate the distance the purse falls during the 1313th second. [2]

  2. Calculate the total distance the purse falls in the first 1313 seconds. [2]

Georgia drops the purse from a height of 12501250 metres above the ground.

  1. Calculate the time, to the nearest second, the purse will take to reach
    the ground. [3]

Georgia visits a national park in Muelle. It is opened at the start of 20192019 and in the first year there were 2000020\hspace{0.15em}000 visitors. The number of people who visit the national park is expected to increase by 8%8\hspace{0.1em}\% each year.

  1. Calculate the number of people expected to visit the national park in 20202020. [2]

  2. Calculate the total number of people expected to visit the national park by the end of 20282028. [3]

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Question 35

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[Maximum mark: 5]

A bouncy ball is dropped out of a second story classroom window, 55\,m off the ground. Every time the ball hits the ground it bounces 8989\,% of its previous height.

  1. Find the height the ball reaches after the 22nd bounce. [2]

  2. Find the total distance the ball has travelled when it hits the ground for the 55th time. [3]

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Question 36

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[Maximum mark: 6]

Let un=4n+1u_n = 4n+1, for nZ+n \in \mathbb{Z}^+.

    1. Using sigma notation, write down an expression for u1+u2+u3++u20u_1 + u_2 + u_3 + \dots + u_{20}.

    2. Find the value of the sum from part (a) (i). [4]

A geometric sequence is defined by vn=9×4n1v_n = 9\times 4^{n-1}, for nZ+n \in \mathbb{Z}^+.

  1. Find the value of the sum of the geometric series k=15vk\medmath{\displaystyle\sum_{k = 1}^5}\hspace{0.1em}v_k.[2]

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Question 37

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[Maximum mark: 6]

Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is 16.116.1 km. The first milestone placed on the road is 124.7124.7 km from London, and the last milestone is near Edinburgh. The length of the road from London to Edinburgh is 667.1667.1 km.

  1. Find the distance between the fifth milestone and London. [3]

  2. Determine how many milestones there are along the road. [3]

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Question 38

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[Maximum mark: 6]

Let un=5n1u_n = 5n-1, for nZ+n \in \mathbb{Z}^+.

    1. Using sigma notation, write down an expression for u1+u2+u3++u10u_1 + u_2 + u_3 + \dots + u_{10}.

    2. Find the value of the sum from part (a) (i). [4]

A geometric sequence is defined by vn=5×2n1v_n = 5\times 2^{n-1}, for nZ+n \in \mathbb{Z}^+.

  1. Find the value of the sum of the geometric series k=16vk\medmath{\displaystyle\sum_{k = 1}^6}\hspace{0.1em}v_k.[2]

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Question 39

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[Maximum mark: 6]

Consider the sum S=k=4l(2k3)\displaystyle S = \sum_{k = 4}^l (2k-3), where ll is a positive integer greater than 44.

  1. Write down the first three terms of the series. [2]

  2. Write down the number of terms in the series. [1]

  3. Given that S=725S = 725, find the value of ll. [3]

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Question 40

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[Maximum mark: 15]

Consider the sequence u1,u2,u3,,un,u_1,\, u_2,\, u_3,\, \dots,\, u_n,\, \dots where

u1=860,u2=980,u3=1100,u4=1220.\begin{aligned} u_1 = 860,\hspace{0.3em} u_2 = 980,\hspace{0.3em} u_3 = 1100,\hspace{0.3em} u_4 = 1220.\end{aligned}

The sequence continues in the same manner.

  1. Find the value of u50u_{50}. [3]

  2. Find the sum of the first 1010 terms of the sequence. [3]

Now consider the sequence v1,v2,v3,,vn,v_1,\, v_2,\, v_3,\, \dots,\, v_n,\, \dots where

v1=2,v2=4,v3=8,v4=16.\begin{aligned} v_1 = 2,\hspace{0.3em} v_2 = 4,\hspace{0.3em} v_3 = 8,\hspace{0.3em} v_4 = 16.\end{aligned}

This sequence continues in the same manner.

  1. Find the exact value of v13v_{13}. [3]

  2. Find the sum of the first 1010 terms of this sequence. [3]

kk is the smallest value of nn for which vnv_n is greater than unu_n.

  1. Calculate the value of kk. [3]

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Question 41

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[Maximum mark: 19]

On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in 2525 seconds. Each lap Eddy cycles takes him 1.61.6 seconds longer than the previous lap.

  1. Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]

Eddy cycles his last lap in 55.455.4 seconds.

  1. Find how many laps he has cycled on Wednesday. [3]

  2. Find the total time, in minutes, cycled by Eddy on Wednesday. [4]

On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in 2525 seconds. Each lap Mario cycles takes him 1.051.05 times as long as his previous lap.

  1. Find the time, in seconds, Mario takes to cycle his fifth lap. [3]

  2. Find the total time, in minutes, Mario takes to cycle his first ten laps. [3]

Each lap Eddy cycles again takes him 1.61.6 seconds longer that his previous lap.
After a certain number of laps Eddy takes less time per lap than Mario.

  1. Find the number of the lap when this happens. [3]

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Question 42

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[Maximum mark: 7]

The half-life, TT, in years, of a radioactive isotope can be modelled by the function

T(k)=ln0.5ln(1k100),0<k<100,\begin{aligned} T(k) &= \dfrac{\ln 0.5}{\ln\left(1 - \frac{k}{100}\right)}, \hspace{0.5em} 0 < k < 100,\end{aligned}

where kk is the decay rate, in percent, per year of the isotope.

  1. The decay rate of Hydrogen-33 is 5.55.5 % per year. Find its half-life.[2]

The half-life of Uranium-232232 (U-232232) is 68.968.9 years. A sample containing 250250 grams of U-232232 is obtained and stored as a side product of a nuclear fuel cycle.

  1. Find the decay rate per year of U-232232. [2]

  2. Find the amount of U-232232 left in the sample after:

    1. 68.968.9 years;

    2. 100100 years. [3]

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Question 43

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[Maximum mark: 7]

The half-life, TT, in days, of a radioactive isotope can be modelled by the function

T(k)=ln0.5ln(1k100),0<k<100,\begin{aligned} T(k) &= \dfrac{\ln 0.5}{\ln\left(1 - \frac{k}{100}\right)}, \hspace{0.5em} 0 < k < 100,\end{aligned}

where kk is the decay rate, in percent, per day of the isotope.

  1. The decay rate of Gold-196196 is 6.26.2 % per day. Find its half-life.[2]

The half-life of Phosphorus-3232 (P-3232) is 14.314.3 days. A sample containing 120120 grams of P-3232 is produced and stored in a biochemistry laboratory.

  1. Find the decay rate per day of P-3232. [2]

  2. Find the amount of P-3232 left in the sample after:

    1. 14.314.3 days;

    2. 5050 days. [3]

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Question 44

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[Maximum mark: 7]

Jenni is conducting an experiment with a spring and has attached a mass so that it will oscillate up and down.

She is measuring the yy-coordinate of the centre of the mass.

At the start of the experiment the mass is at rest with its centre being at the point (0,0)(0, 0).

She gives the mass a nudge upwards in the positive yy-direction. She makes her first measurement of (0,37.5)(0, 37.5) when the centre of the mass is at the first maximum point (n=1n=1). The units of the yy-coordinate are in millimetres.

The mass then moves downwards passing the xx-axis and reaching its first minimum point (n=2n=2). Jenni makes her second measurement of the yy-coordinate of the centre of a the mass as (0,a)(0, a).

The mass then moves up past the xx-axis to the next maximum point (n=3n=3) and Jenni makes her third measurement of (0,24)(0, 24).

The diagram below shows how the mass moves up and down until Jenni makes her 33rd measurement.

springs

Jenni notices that the yy-coordinates of the three measurements 37.5,  a,  2437.5,\; a,\; 24 form a geometric sequence.

  1. Find aa. [2]

The spring continues to oscillate up and down with Jenni measuring the yy-coordinate in the same way as described.

The results continue to form a geometric sequence.

  1. Find the 66th term in the sequence. Give your answer to 3 decimal places. [2]

  2. Show that the total distance travelled in the yy-direction by the mass when the 66th measurement is made is 264.408264.408 mm. [3]

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