A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 3: Sequences and series · Lesson 2

Geometric sequences and series

113 marks

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Date

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Quick questions to start. They use what you need today.

A1
Needed today
Solve $3 \times 2^{x} = 600$ , giving $x$ to $3$ significant figures.
[2 marks]
Answer
A2
Common slip
Find the smallest integer $n$ for which $0. 8^{n} < 0.05$ .
[3 marks]
Answer
A3
Needed today
A geometric sequence has first term $6$ and common ratio $- \frac{1}{2}$ . Find the 4th term.
[2 marks]
Answer
A4
Needed today
In a geometric sequence $u_{2} = 12$ and $u_{5} = - 96$ . Find the common ratio.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

In a geometric sequence $u_{3} = 18$ and $u_{6} = 486$ . Find the first term, the common ratio and $u_{10}$ .

Hint Divide the two terms:

\frac{a r ^{5}}{a r ^{2}} = r^{3}

B1
In a geometric sequence $u_{2} = 40$ and $u_{5} = \frac{5}{8}$ . Find the first term, the common ratio and $u_{8}$ as an exact fraction.
[4 marks]
Answer

Example 2

In a geometric series $u_{3} = 36$ and $u_{5} = 81$ . Find the two possible values of $S_{6}$ .

Hint

r^{2} = \frac{81}{36}

has two square roots. Work out

a

and

S_{6}

for each one.

B2
In a geometric series $u_{2} = - 12$ and $u_{4} = - 3$ . Find the two possible values of $S_{5}$ .
[5 marks]
Answer

Example 3

Prove that the sum of the first $n$ terms of a geometric series with first term $a$ and common ratio $r$ , $r \neq = 1$ , is $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ .

Hint Write

S_{n}

and

r S_{n}

underneath it. Subtract: almost every term cancels.

B3
Prove that $1 + 2 + 4 + \dots + 2^{n - 1} = 2^{n} - 1$ .
[4 marks]

Example 4

Find the least number of terms of $4 + 6 + 9 + \dots$ for which the sum is greater than $1000$ .

Hint Write

S_{n}

with

r = 1.5

, make

1. 5^{n}

the subject, then take logs.

B4
The geometric sequence $200, 180, 162, \dots$ has $n$ th term $u_{n}$ . Find the first term that is less than $20$ , and its value.
[5 marks]
Answer

Example 5

The sum of the first $n$ terms of $3 + 6 + 12 + \dots$ is $3069$ . Find $n$ .

Hint Set the sum formula equal to

3069

and make

2^{n}

the subject.

B5
The sum of the first $n$ terms of $5 - 10 + 20 - 40 + \dots$ is $- 1705$ . Find $n$ .
[5 marks]
Answer

Practice

A-Level

C1
A student wants the least $n$ for which $0.7 5^{n} < 0.01$ . She writes that $n lo g 0.75 < lo g 0.01$ , so $n < \frac{lo g 0.01}{lo g 0.75} = 16.01$ , and so $n = 16$ . Explain her mistake and find the correct value of $n$ .
[4 marks]
Answer $n = \dots\dots$

C2
The expressions $(2 k + 3)$ , $(k + 6)$ and $k$ are consecutive terms of a geometric sequence. Find the two possible values of $k$ and the common ratio in each case.
[5 marks]
Answer
C3
Find the sum of the first $8$ terms of $2 - 6 + 18 - 54 + \dots$ .
[4 marks]
Answer
C4
For the series $3 + 6 + 12 + \dots$ , a student finds $u_{5} + u_{6} + \dots + u_{12}$ by working out $S_{12} - S_{5}$ . Explain why this is wrong, and find the correct value.
[4 marks]
Answer
C5
A geometric series has first term $5$ , and the sum of its first $3$ terms is $65$ . Find the possible values of the common ratio.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A geometric series has second term $50$ and fifth term $25.6$ .
(a)
Find the common ratio and the first term.
[3 marks]
Answer
(b)
Find the sum of the first $10$ terms, to $1$ decimal place.
[2 marks]
Answer
(c)
Find the least value of $n$ for which $S_{n} > 300$ .
[4 marks]
Answer
Total for D1: 9 marks

D2
A geometric sequence has $u_{4} = 16$ and $u_{6} = 64$ . Kai writes: " $r^{2} = 4$ , so $r = 2$ and $a = 2$ . There is only one sequence."
Kai is wrong.
(a)
Explain Kai's mistake, and give every possible pair of values of $r$ and $a$ .
[3 marks]
(b)
Find $S_{8}$ for each pair.
[3 marks]
Answer
(c)
For the series in which every term is positive, find the least $n$ for which $S_{n} > 1 0^{6}$ .
[3 marks]
Answer
Total for D2: 9 marks
D3
A geometric series has positive terms. The sum of the first $2$ terms is $16$ and the sum of the first $4$ terms is $80$ . Find the sum of the first $10$ terms.
[7 marks]
Answer
D4
The first three terms of a geometric sequence are $(x + 4)$ , $x$ and $(x - 3)$ .
(a)
Find the value of $x$ .
[3 marks]
Answer
(b)
Find the 10th term as an exact fraction.
[3 marks]
Answer
(c)
Find the least $n$ for which the sum of the first $n$ terms is greater than $63$ .
[4 marks]
Answer
Total for D4: 10 marks
D5
A geometric series has first term $a$ and common ratio $r$ , where $r \neq = 1$ .
(a)
Prove that the sum of the first $n$ terms is $\frac{a ( 1 - r ^{n} )}{1 - r}$ .
[4 marks]
(b)
Hence show that $1 + 3 + 9 + \dots + 3^{n - 1} = \frac{3 ^{n} - 1}{2}$ , and find $n$ when this sum is $9841$ .
[5 marks]
Answer
Total for D5: 9 marks
D6
The sum of the first $n$ terms of a series is $S_{n} = 2 (3^{n} - 1)$ . Prove that the series is geometric, and find $u_{11} + u_{12} + \dots + u_{20}$ in the form $k \times 3^{m} (3^{m} - 1)$ , where $k$ and $m$ are integers.
[7 marks]
Answer $k = \dots\dots$ , $m = \dots\dots$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The numbers $a$ , $b$ and $c$ are three different consecutive terms of a geometric sequence. The numbers $a$ , $2 b$ and $3 c$ are consecutive terms of an arithmetic sequence. Given that $a + b + c = 26$ , find $a$ , $b$ and $c$ .
[9 marks]
Hint 1 · What to try
Let $b = a r$ and $c = a r^{2}$ .
Hint 2 · The first line
In an arithmetic sequence the middle term is the mean of the other two, so $2 (2 b) = a + 3 c$ .
Hint 3 · The full method
You should get $3 r^{2} - 4 r + 1 = 0$ . One root gives three equal numbers.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the first term and common ratio from two terms, including a negative ratio.
I can prove and use the formula for the sum of a geometric series.
I can use logarithms to find the least $n$ for a condition, turning the inequality round when I divide by a negative log.
I can solve problems where three expressions are consecutive terms.

Answers and mark scheme

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