A-Level Maths | Sequences & Series

Sequences

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) $u_{n+1} = \dfrac{1}{2 + u_n}, \quad n \geq 1, \quad u_1 = 0$

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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

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

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

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

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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.

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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.

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a) $a = 43$, $d = -3$    b) $15$    c) $S_{15} = 330$

Geometric Sequences

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

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

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

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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) $\displaystyle\sum_{r=3}^{9} \left[2 \times (-3)^r\right]$

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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.

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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.

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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 $30\,000$.

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

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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.

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a) $k = 15$    b) $S_\infty = 62.5$