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# A-Level Maths | Sequences & Series
- URL: https://www.a-level-maths-tutor.com/a-level-maths-sequences-series/
- Published: 2026-06-13T10:41:59.000Z
- Updated: 2026-07-29T12:19:58.000Z
- Author: Jaisul Naik
- Tags: Pure Maths

### Sequences

$n$: the position of a term · $a$: first term · $d$: common difference · $r$: common ratio

## Question 1

Write down the first four terms of each sequence.

**a)** $u\_n = u\_{n-1} + 4, \\quad n > 1, \\quad u\_1 = 3$

**b)** $u\_n = 3u\_{n-1} + 1, \\quad n > 1, \\quad u\_1 = 2$

**c)** $u\_{n+1} = 2u\_n + 5, \\quad n > 0, \\quad u\_1 = -2$

**d)** $u\_n = 7 - u\_{n-1}, \\quad n \\geq 2, \\quad u\_1 = 5$

**e)** $u\_n = 2(5 - 2u\_{n-1}), \\quad n > 1, \\quad u\_1 = -1$

**f)** $u\_n = \\tfrac{1}{10}(u\_{n-1} + 20), \\quad n \\geq 2, \\quad u\_1 = 10$

**g)** $u\_{n+1} = 1 - \\tfrac{1}{3}u\_n, \\quad n \\geq 1, \\quad u\_1 = 6$

**h)** $u\_{n+1} = \\dfrac{1}{2 + u\_n}, \\quad n \\geq 1, \\quad u\_1 = 0$

Show Answers 

**a)** $3, 7, 11, 15$ **b)** $2, 7, 22, 67$ **c)** $-2, 1, 7, 19$ **d)** $5, 2, 5, 2$

**e)** $-1, 14, -46, 194$ **f)** $10, 3, 2.3, 2.23$ **g)** $6, -1, \\tfrac{4}{3}, \\tfrac{5}{9}$ **h)** $0, \\tfrac{1}{2}, \\tfrac{2}{5}, \\tfrac{5}{12}$

### Arithmetic Sequences

$n$th term: $a + (n-1)d$

Sum of first $n$ terms: $S\_n = \\tfrac{1}{2}n\\left\[2a + (n-1)d\\right\]$

## Question 2

The $n$th term of each of the following sequences is given by $u\_n = an + b$, for $n \\geq 1$. Find the values of the constants $a$ and $b$ in each case.

**a)** $4, 7, 10, 13, 16, \\ldots$

**b)** $0, 7, 14, 21, 28, \\ldots$

**c)** $16, 14, 12, 10, 8, \\ldots$

**d)** $0.4, 1.7, 3.0, 4.3, 5.6, \\ldots$

**e)** $100, 83, 66, 49, 32, \\ldots$

**f)** $-13, -5, 3, 11, 19, \\ldots$

Show Answers 

**a)** $a = 3,\\ b = 1$ **b)** $a = 7,\\ b = -7$ **c)** $a = -2,\\ b = 18$

**d)** $a = 1.3,\\ b = -0.9$ **e)** $a = -17,\\ b = 117$ **f)** $a = 8,\\ b = -21$

## Question 3

For each of the following arithmetic series, write down the common difference and find the value of the 40th term.

**a)** $4, 10, 16, 22, \\ldots$

**b)** $30, 27, 24, 21, \\ldots$

**c)** $8.9, 11.2, 13.5, 15.8, \\ldots$

Show Answers 

**a)** $d = 6$, $u\_{40} = 238$ **b)** $d = -3$, $u\_{40} = -87$ **c)** $d = 2.3$, $u\_{40} = 98.6$

## Question 4

Find the sum of the first 30 terms of each of the following arithmetic series.

**a)** $8, 12, 16, 20, \\ldots$

**b)** $60, 53, 46, 39, \\ldots$

**c)** $7\\tfrac{1}{4}, 8\\tfrac{3}{4}, 10\\tfrac{1}{4}, 11\\tfrac{3}{4}, \\ldots$

Show Answers 

**a)** $1980$ **b)** $-1245$ **c)** $870$

## Question 5

The first and third terms of an arithmetic series are $21$ and $27$ respectively.

**a)** Find the common difference of the series.

**b)** Find the sum of the first 40 terms of the series.

Show Answers 

**a)** $d = 3$ **b)** $S\_{40} = 3180$

## Question 6

The sum of the first six terms of an arithmetic series is $213$ and the sum of the first ten terms of the series is $295$.

**a)** Find the first term and common difference of the series.

**b)** Find the number of positive terms in the series.

**c)** Hence find the maximum value of $S\_n$, the sum of the first $n$ terms of the series.

Show Answers 

**a)** $a = 43$, $d = -3$ **b)** $15$ **c)** $S\_{15} = 330$

### Geometric Sequences

$n$th term: $ar^{n-1}$

Sum of first $n$ terms: $S\_n = \\dfrac{a(1 - r^n)}{1 - r}$ or $\\dfrac{a(r^n - 1)}{r - 1}$

Sum to infinity (when $|r| < 1$): $S\_\\infty = \\dfrac{a}{1 - r}$

## Question 7

For each of the following geometric series, write down the common ratio and find the value of the eighth term.

**a)** $3, 9, 27, 81, \\ldots$

**b)** $1024, 256, 64, 16, \\ldots$

**c)** $1, -2, 4, -8, \\ldots$

Show Answers 

**a)** $r = 3$, $u\_8 = 6561$ **b)** $r = \\tfrac{1}{4}$, $u\_8 = \\tfrac{1}{16}$ **c)** $r = -2$, $u\_8 = -128$

## Question 8

For each of the following geometric series, find an expression for the $n$th term.

**a)** $1, 5, 25, 125, \\ldots$

**b)** $3, -12, 48, -192, \\ldots$

**c)** $81, 54, 36, 24, \\ldots$

Show Answers 

**a)** $u\_n = 5^{n-1}$ **b)** $u\_n = 3 \\times (-4)^{n-1}$ **c)** $u\_n = 81 \\times \\left(\\tfrac{2}{3}\\right)^{n-1}$

## Question 9

Given the first term $a$, the common ratio $r$, and the number of terms $n$, find the sum of each of the following geometric series. Give your answers to 3 decimal places where appropriate.

**a)** $a = 4,\\ r = 3,\\ n = 8$

**c)** $a = -1,\\ r = -4,\\ n = 12$

**d)** $a = 200,\\ r = 0.7,\\ n = 20$

Show Answers 

**a)** $13\\,120$ **c)** $3\\,355\\,443$ **d)** $666.135$

## Question 10

Evaluate to an appropriate degree of accuracy.

**a)** $\\displaystyle\\sum\_{r=1}^{9} 3^r$ **c)** $\\displaystyle\\sum\_{r=1}^{10} (10 \\times 2^r)$ **e)** $\\displaystyle\\sum\_{r=1}^{10} \\left\[12 \\times \\left(\\tfrac{1}{6}\\right)^r\\right\]$ **h)** $\\displaystyle\\sum\_{r=3}^{9} \\left\[2 \\times (-3)^r\\right\]$

Show Answers 

**a)** $29\\,523$ **c)** $20\\,460$ **e)** $2.400$ **h)** $-29\\,538$

## Question 11

The second and third terms of a geometric series are $2$ and $10$ respectively.

**a)** Find the common ratio of the series.

**b)** Find the first term of the series.

**c)** Find the sum of the first eight terms of the series.

Show Answers 

**a)** $r = 5$ **b)** $a = 0.4$ **c)** $S\_8 = 39\\,062.4$

## Question 12

The first and fourth terms of a geometric series are $2$ and $54$ respectively.

**a)** Find the common ratio of the series.

**b)** Find the ninth term of the series.

Show Answers 

**a)** $r = 3$ **b)** $u\_9 = 13\\,122$

## Question 13

The sum of the first four terms of a geometric series is $130$ and its common ratio is $1\\tfrac{1}{2}$.

**a)** Find the first term of the series.

**b)** Find the eighth term of the series.

**c)** Find the least value of $n$ for which the sum of the first $n$ terms of the series is greater than $30\\,000$.

Show Answers 

**a)** $a = 16$ **b)** $u\_8 = 273\\tfrac{3}{8}$ **c)** $n = 17$

## Question 14

For each of the following geometric series, either find its sum to infinity or explain why this cannot be found.

**a)** $12, 6, 3, 1.5, \\ldots$

**b)** $270, 90, 30, 10, \\ldots$

**d)** $216, 144, 96, 64, \\ldots$

**f)** $500, -300, 180, -108, \\ldots$

Show Answers 

**a)** $24$ **b)** $405$ **d)** $648$ **f)** $312.5$

## Question 15

The first three terms of a geometric series are $(k + 10)$, $k$ and $(k - 6)$ respectively.

**a)** Find the value of the constant $k$.

**b)** Find the sum to infinity of the series.

Show Answers 

**a)** $k = 15$ **b)** $S\_\\infty = 62.5$

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