Sequences & Series

[Maximum mark: 6]

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

1. Find the common difference, $d$. [2]
   

2. Find the $10$th term in the sequence. [2]
   

3. Find the sum of the first $10$ terms in the sequence. [2]
   

[Maximum mark: 6]

An arithmetic sequence has $u_1= 40$, $u_2 = 32$, $u_3 = 24$.

1. Find the common difference, $d$. [2]
   

2. Find $u_8$. [2]
   

3. Find $S_8$. [2]
   

[Maximum mark: 6]

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

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]
   

[Maximum mark: 6]

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

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]
   

[Maximum mark: 6]

Consider the infinite geometric sequence $4480$, $-3360$, $2520$, $-1890,\dots$

1. Find the common ratio, $r$. [2]
   

2. Find the $20$th term. [2]
   

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

[Maximum mark: 6]

The table shows the first four terms of three sequences: $u_n$, $v_n$, and $w_n$.

1. State which sequence is

   1. arithmetic;

   2. geometric. [2]
      

2. Find the sum of the first $50$ terms of the arithmetic sequence. [2]
   

3. Find the exact value of the $13$th term of the geometric sequence. [2]
   

[Maximum mark: 7]

An arithmetic sequence is given by $3$, $5$, $7,\dots$

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

2. Find

   1. $u_{10}$;

   2. $S_{10}$. [4]
      

3. Given that $u_n = 253$, find the value of $n$. [2]
   

[Maximum mark: 6]

Consider the infinite geometric sequence $9000$, $-7200$, $5760$, $-4608$, ...

1. Find the common ratio. [2]
   

2. Find the $25$th term. [2]
   

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

[Maximum mark: 6]

A tennis ball bounces on the ground $n$ times. The heights of the bounces, $h_1, h_2, h_3, \dots,h_n,$ form a geometric sequence. The height that the ball bounces the first time, $h_1$, is $80$ cm, and the second time, $h_2$, is $60$ cm.

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

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

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

[Maximum mark: 6]

The third term, $u_3$, of an arithmetic sequence is $7$. The common difference of
the sequence, $d$, is $3$.

1. Find $u_1$, the first term of the sequence. [2]
   

2. Find $u_{60}$, the $60$th term of sequence. [2]
   

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

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

[Maximum mark: 6]

The fifth term, $u_5$, of a geometric sequence is $125$. The sixth term, $u_6$, is $156.25$.

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

2. Find $u_1$, the first term of the sequence. [2]
   

3. Calculate the sum of the first $12$ terms of the sequence. [2]
   

[Maximum mark: 6]

The fourth term, $u_4$, of a geometric sequence is $135$. The fifth term, $u_5$, is $81$.

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

2. Find $u_1$, the first term of the sequence. [2]
   

3. Calculate the sum of the first $20$ terms of the sequence. [2]
   

[Maximum mark: 6]

The fifth term, $u_5$, of an arithmetic sequence is $25$. The eleventh term, $u_{11}$, of the same sequence is $49$.

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

2. Find $u_1$, the first term of the sequence. [2]
   

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

[Maximum mark: 6]

Consider the following sequence of figures.

Figure 1 contains $6$ line segments.

1. Given that Figure $n$ contains $101$ line segments, show that $n = 20$.[3]
   

2. Find the total number of line segments in the first $20$ figures. [3]
   

[Maximum mark: 5]

Consider an arithmetic sequence where $u_{12} = S_{12} = 12$. Find the value of the first term, $u_1$, and the value of the common difference, $d$.

[Maximum mark: 6]

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

1. Find the common difference. [2]
   

2. Find the first term. [2]
   

3. Find the sum of the first $20$ terms in the sequence. [2]
   

[Maximum mark: 6]

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

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

2. Find $u_6$. [2]
   

3. Find $S_{\infty}$. [2]
   

[Maximum mark: 6]

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

1. Find the common difference. [2]
   

2. Find the first term. [2]
   

3. Find the sum of the first $11$ terms in the sequence. [2]
   

[Maximum mark: 5]

In an arithmetic sequence, the sum of the 2nd and 6th term is $32$.
Given that the sum of the first six terms is $120$, determine the first term and common difference of the sequence.

[Maximum mark: 5]

An arithmetic sequence has first term $45$ and common difference $-1.5$.

1. Given that the $k$th term of the sequence is zero, find the value of $k$. [2]
   

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

2. Find the maximum value of $S_n$. [3]
   

[Maximum mark: 6]

An arithmetic sequence has first term $-30$ and common difference $5$.

1. Given that the $k$th term is the first positive term of the sequence, find the value of $k$. [3]
   

Let $S_n$ denote the sum of the first $n$ terms of the sequence.

2. Find the minimum value of $S_n$. [3]
   

[Maximum mark: 6]

The Australian Koala Foundation estimates that there are about $45\hspace{0.15em}000$ koalas left in the wild in $2019$. A year before, in $2018$, the population of koalas was estimated as $50\hspace{0.15em}000$. Assuming the population of koalas continues to decrease by the same percentage each year, find:

1. the exact population of koalas in $2022$; [3]
   

2. the number of years it will take for the koala population to reduce to half of its number in $2018$. [3]
   

[Maximum mark: 6]

Landmarks are placed along the road from London to Edinburgh and the distance between each landmark is $16.1$ km. The first landmark placed on the road is $124.7$ km from London, and the last landmark is near Edinburgh. The length of the road from London to Edinburgh is $667.1$ km.

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

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

[Maximum mark: 6]

The first term of an arithmetic sequence is $24$ and the common difference is $16$.

1. Find the value of the $62$nd term of the sequence. [2]
   

The first term of a geometric sequence is $8$. The $4$th term of the geometric sequence is equal to the $13$th term of the arithmetic sequence given above.

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

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

[Maximum mark: 6]

On $1$st of January $2021$, Fiona decides to take out a bank loan to purchase a new Tesla electric car. Fiona takes out a loan of $\$P$ with a bank that offers a nominal annual interest rate of $2.6\hspace{0.05em}\%$, compounded monthly.

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

1. Find the value of $\alpha$, 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.

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

[Maximum mark: 6]

The first three terms of an arithmetic sequence are $u_1, 4u_1-9$, and $3u_1+18$.

1. Show that $u_1=9$. [2]
   

2. Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number. [4]
   

[Maximum mark: 6]

On Gary's $50$th birthday, he invests $\$P$ in an account that pays a nominal annual interest rate of $5$ %, compounded monthly.

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

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

Gary makes no further deposits or withdrawals from the account.

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

[Maximum mark: 7]

In an arithmetic sequence, the third term is $41$ and the ninth term is $23$.

1. Find the common difference. [2]
   

2. Find the first term. [2]
   

3. Find the smallest value of $n$ such that $S_n < 0$. [3]
   

[Maximum mark: 7]

The first three terms of a geometric sequence are $u_1 = 0.8$, $u_2 = 2.4$, $u_3 = 7.2$.

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

2. Find the value of $S_8$. [2]
   

3. Find the least value of $n$ such that $S_n > 35\hspace{0.15em}000$. [3]
   

[Maximum mark: 7]

The first three terms of a geometric sequence are $u_1 = 0.4$, $u_2 = 0.6$, $u_3 = 0.9$.

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

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

3. Find the greatest value of $n$ such that $S_n < 650$. [3]
   

[Maximum mark: 7]

In a geometric sequence, $u_2 = 6$, $u_5 = 20.25$.

1. Find the common ratio, $r$. [2]
   

2. Find $u_1$. [2]
   

3. Find the greatest value of $n$ such that $u_n < 200$. [3]
   

[Maximum mark: 6]

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

A population of goats on an island starts at $232$. The population is expected to increase by $15$ % each year.

1. Find the expected population size after:

   1. $10$ years;

   2. $20$ years. [4]
      

2. Find the number of years it will take for the population to reach $15\hspace{0.15em}000$. [2]
   

[Maximum mark: 5]

Consider an arithmetic sequence with $u_{1}=5$ and $u_{6}=\log_3 32$.

Find the common difference of the sequence, expressing your answer in the form $\log_3 a$, where $a \in \mathbb{Q}$.

[Maximum mark: 6]

On $1$st of January $2022$, Grace invests $\$P$ in an account that pays a nominal annual interest rate of $6$ %, compounded quarterly.

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

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

Grace makes no further deposits or withdrawals from the account.

2. Find the year in which the amount of money in Grace's account will become triple the amount she invested. [3]
   

[Maximum mark: 6]

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

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 $S = 725$, find the value of $l$. [3]
   

[Maximum mark: 6]

Let $u_n = 5n-1$, for $n \in \mathbb{Z}^+$.

1. 1. Using sigma notation, write down an expression for $u_1 + u_2 + u_3 + \dots + u_{10}$.

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

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

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