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IB Mathematics AA SL - Questionbank

Sequences & Series

Arithmetic/Geometric, Sigma Notation, Applications, Compound Interest…

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Question 1

calculator

easy

[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]

easy

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Question 2

no calculator

easy

[Maximum mark: 6]

Consider an arithmetic sequence 2,6,10,14,2,6,10,14,\dots

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

  2. Find the 1010th term in the sequence. [2]

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

easy

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Question 3

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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]

easy

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Question 4

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easy

[Maximum mark: 6]

Consider the infinite geometric sequence 44804480, 3360-3360, 25202520, 1890,-1890,\dots

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

  2. Find the 2020th term. [2]

  3. Find the exact sum of the infinite sequence. [2]

easy

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Question 5

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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]

easy

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Question 6

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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]

easy

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Question 7

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easy

[Maximum mark: 6]

Jeremy invests $8000\$8000 into a savings account that pays an annual interest rate of 5.55.5 %, compounded annually.

  1. Write down a formula which calculates that total value of the investment after nn years. [2]

  2. Calculate the amount of money in the savings account after:

    1. 11 year;

    2. 33 years. [2]

  3. Jeremy wants to use the money to put down a $10000\$10\hspace{0.15em}000 deposit on an apartment. Determine if Jeremy will be able to do this within a 55-year timeframe.[2]

easy

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Question 8

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easy

[Maximum mark: 6]

Consider the infinite geometric sequence 90009000, 7200-7200, 57605760, 4608-4608, ...

  1. Find the common ratio. [2]

  2. Find the 2525th term. [2]

  3. Find the exact sum of the infinite sequence. [2]

easy

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Question 9

no calculator

easy

[Maximum mark: 7]

An arithmetic sequence is given by 33, 55, 7,7,\dots

  1. Write down the value of the common difference, dd. [1]

  2. Find

    1. u10u_{10};

    2. S10S_{10}. [4]

  3. Given that un=253u_n = 253, find the value of nn. [2]

easy

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Question 10

no calculator

easy

[Maximum mark: 6]

Consider the following sequence of figures.

AA008

Figure 1 contains 66 line segments.

  1. Given that Figure nn contains 101101 line segments, show that n=20n = 20.[3]

  2. Find the total number of line segments in the first 2020 figures. [3]

easy

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Question 11

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easy

[Maximum mark: 6]

Hannah buys a car for $24900\$24\hspace{0.15em}900. The value of the car depreciates by 1616 % each year.

  1. Find the value of the car after 1010 years. [3]

Patrick buys a car for $12000\$12\hspace{0.15em}000. The car depreciates by a fixed percentage each year, and after 66 years it is worth $6200\$6200.

  1. Find the annual rate of depreciation of the car. [3]

easy

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Question 12

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easy

[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.

AA724a

  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]

easy

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Question 13

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easy

[Maximum mark: 6]

Julia wants to buy a house that requires a deposit of 7400074\hspace{0.15em}000 Australian dollars (AUD).

Julia is going to invest an amount of AUD in an account that pays a nominal annual interest rate of 5.55.5 %, compounded monthly.

  1. Find the amount of AUD Julia needs to invest to reach 7400074\hspace{0.15em}000 AUD after 88 years. Give your answer correct to the nearest dollar. [3]

Julia's parents offer to add 50005000 AUD to her initial investment from part (a), however, only if she invests her money in a more reliable bank that pays a nominal annual interest rate only of 3.53.5 %, compounded quarterly.

  1. Find the number of years it would take Julia to save the 7400074\hspace{0.15em}000 AUD if she accepts her parents money and follows their advice. Give your answer correct to the nearest year. [3]

easy

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Question 14

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easy

[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]

easy

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Question 15

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easy

[Maximum mark: 6]

In an arithmetic sequence, u5=24u_5 = 24, u13=80u_{13} = 80.

  1. Find the common difference. [2]

  2. Find the first term. [2]

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

easy

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Question 16

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easy

[Maximum mark: 6]

In this question give all answers correct to two decimal places.

Mia deposits 40004000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 66 %, compounded semi-annually.

  1. Find the amount of interest that Mia will earn over the next 2.52.5 years. [3]

Ella also deposits AUD into a bank account. Her bank pays a nominal annual interest\text{interest} rate of 44 %, compounded monthly. In 2.52.5 years, the total amount in Ella's account will be 40004000 AUD.

  1. Find the amount that Ella deposits in the bank account. [3]

easy

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Question 17

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easy

[Maximum mark: 5]

Maria invests $25000\$25\hspace{0.15em}000 into a savings account that pays a nominal annual interest rate of 4.254.25%, compounded monthly.

  1. Calculate the amount of money in the savings account after 33 years. [2]

  2. Calculate the number of years it takes for the account to reach $40000\$40\hspace{0.15em}000. [3]

easy

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Question 18

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easy

[Maximum mark: 6]

The first three terms of a geometric sequence are u1=32u_1 = 32, u2=16u_2 = -16, u3=8u_3 = 8.

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

  2. Find u6u_6. [2]

  3. Find SS_{\infty}. [2]

easy

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Question 19

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easy

[Maximum mark: 6]

In an arithmetic sequence, u4=12u_4 = 12, u11=9u_{11} = -9.

  1. Find the common difference. [2]

  2. Find the first term. [2]

  3. Find the sum of the first 1111 terms in the sequence. [2]

easy

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Question 20

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easy

[Maximum mark: 6]

Emily deposits 20002000 Australian dollars (AUD) into a bank account. The bank pays a nominal annual interest rate of 44 %, compounded monthly.

  1. Find the amount of money that Emily will have in her bank account after 55 years. Give your answer correct to two decimal places. [3]

Emily will withdraw the money back from her bank account when the amount reaches 30003000 AUD.

  1. Find the time, in months, until Emily withdraws the money from her bank account. [3]

easy

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Question 21

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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]

easy

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Question 22

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easy

[Maximum mark: 5]

In an arithmetic sequence, the sum of the 2nd and 6th term is 3232.
Given that the sum of the first six terms is 120120, determine the first term and common difference of the sequence.

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Question 23

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easy

[Maximum mark: 6]

Ali bought a car for $18000\$18\hspace{0.15em}000. The value of the car depreciates by 10.510.5 % each year.

  1. Find the value of the car at the end of the first year. [2]

  2. Find the value of the car after 44 years. [2]

  3. Calculate the number of years it will take for the car to be worth exactly half its original value. [2]

easy

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Question 24

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easy

[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]

easy

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Question 25

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easy

[Maximum mark: 5]

An arithmetic sequence has first term 4545 and common difference 1.5-1.5.

  1. Given that the kkth term of the sequence is zero, find the value of kk. [2]

Let SnS_n denote the sum of the first nn terms of the sequence.

  1. Find the maximum value of SnS_n. [3]

easy

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Question 26

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easy

[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]

easy

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Question 27

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easy

[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]

easy

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Question 28

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easy

[Maximum mark: 6]

Greg has saved 20002000 British pounds (GBP) over the last six months. He decided to deposit his savings in a bank which offers a nominal annual interest rate of 8%\text{\(8\)\hspace{0.05em}\%}, compounded monthly, for two years.

  1. Calculate the total amount of interest Greg would earn over the two years. Give your answer correct to two decimal places. [3]

Greg would earn the same amount of interest, compounded semi-annually, for two years if he deposits his savings in a second bank.

  1. Calculate the nominal annual interest rate the second bank offers. [3]

easy

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Question 29

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medium

[Maximum mark: 6]

On 11st of January 20212021, Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a loan of $P\$P with a bank that offers a nominal annual interest rate of 2.6%2.6\hspace{0.05em}\%, compounded monthly.

The size of Fiona's loan at the end of each year follows a geometric sequence with common ratio, α\alpha.

  1. Find the value of α\alpha, giving your answer to five significant figures. [3]

The bank lets the size of Fiona's loan increase until it becomes triple the size of the original loan. Once this happens, the bank demands that Fiona pays the entire amount back to close the loan.

  1. Find the year during which Fiona will need to pay back the loan. [3]

medium

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Question 30

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medium

[Maximum mark: 6]

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

  1. Find the value of the 6262nd 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 31

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medium

[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 32

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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 landmark placed on the road is 124.7124.7 km from London, and the last landmark is near Edinburgh. The length of the road from London to Edinburgh is 667.1667.1 km.

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

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

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Question 33

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medium

[Maximum mark: 6]

On Gary's 5050th birthday, he invests $P\$P in an account that pays a nominal annual interest rate of 55 %, compounded monthly.

The amount of money in Gary's account at the end of each year follows a geometric sequence with common ratio, α\alpha.

  1. Find the value of α\alpha, giving your answer to four significant figures. [3]

Gary makes no further deposits or withdrawals from the account.

  1. Find the age Gary will be when the amount of money in his account will be double the amount he invested. [3]

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Question 34

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medium

[Maximum mark: 6]

The fourth term of an arithmetic sequence is equal to 1313 and the sum of the first 1010 terms is 5555.

  1. Find the common difference and the first term. [4]

  2. Determine the greatest value of nn such that the nnth term is positive. [2]

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