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

Sum to infinity

114 marks

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

A1
From chapter 3
A geometric series has first term $12$ and common ratio $\frac{1}{3}$ . Find $S_{4}$ as an exact fraction.
[2 marks]
Answer
A2
Needed today
Solve $∣2 x - 1∣ < 1$ .
[2 marks]
Answer
A3
Common slip
Does the series $5 - 7.5 + 11.25 - \dots$ have a sum to infinity? Give a reason.
[2 marks]
Answer
A4
Needed today
Write $0. 4 ˙ 5 ˙$ as a fraction in its simplest form.
[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

Find the sum to infinity of $24 + 18 + 13.5 + \dots$ .

Hint Find

r

by dividing a term by the one before. Check

∣ r ∣ < 1

before using the formula.

B1
Find the sum to infinity of $50 - 20 + 8 - \dots$ .
[4 marks]
Answer

Example 2

A geometric series has first term $12$ and sum to infinity $48$ . Find the common ratio.

Hint Put the two numbers into

S_{\infty} = \frac{a}{1 - r}

and solve for

r

B2
A geometric series has first term $15$ and sum to infinity $9$ . Find the common ratio and the third term.
[4 marks]
Answer

Example 3

A geometric series has second term $8$ and sum to infinity $36$ . Find the two possible values of the common ratio, and the first term in each case.

Hint Write

a r = 8

and

\frac{a}{1 - r} = 36

. Replace

a

with

\frac{8}{r}

to get a quadratic in

r

B3
A geometric series has second term $6$ and sum to infinity $32$ . Find the two possible values of the common ratio, and the first term in each case.
[6 marks]
Answer

Example 4

The series $1 + (2 x - 1) + (2 x - 1)^{2} + \dots$ has a sum to infinity. Find the range of values of $x$ , and write the sum to infinity in terms of $x$ .

Hint Here

r = 2 x - 1

. The series converges when

∣ r ∣ < 1

B4
A geometric series has first term $2$ and common ratio $\frac{x - 3}{2}$ . Find the range of values of $x$ for which it has a sum to infinity, and find $x$ when the sum to infinity is $8$ .
[5 marks]
Answer

Example 5

A geometric series has first term $20$ and common ratio $0.9$ . Find the least $n$ for which $S_{\infty} - S_{n} < 1$ .

Hint

S_{\infty} - S_{n} = \frac{a r ^{n}}{1 - r}

. Make

0. 9^{n}

the subject and take logs. Turn the inequality round when you divide by

lo g 0.9

B5
A geometric series has first term $60$ and common ratio $0.6$ . Find the least $n$ for which $S_{n} > 149.9$ .
[5 marks]
Answer

Practice

A-Level

C1
Sam writes: "The sum to infinity of $6 + 9 + 13.5 + \dots$ is $\frac{6}{1 - 1.5} = - 12$ ." Explain why Sam is wrong. Then find the set of values of $k$ for which the series $6 + 6 k + 6 k^{2} + \dots$ has a sum to infinity greater than $12$ .
[5 marks]
Answer $\dots\dots < k < \dots\dots$

C2
Use a geometric series to write $0.2 3 ˙ 6 ˙$ as a fraction in its simplest form.
[5 marks]
Answer
C3
A geometric series has first term $81$ and common ratio $- \frac{1}{3}$ . Find its sum to infinity, and the sum to infinity of its positive terms only.
[5 marks]
Answer
C4
A geometric series has first term $9$ and third term $4$ . Find the two possible values of its sum to infinity.
[4 marks]
Answer
C5
The sum to infinity of a geometric series is four times its first term. The third term is $18$ . Find the common ratio and the sum to infinity.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A geometric series has second term $12$ and sum to infinity $75$ .
(a)
Show that the common ratio $r$ satisfies $25 r^{2} - 25 r + 4 = 0$ .
[3 marks]
(b)
Find the two possible values of $r$ .
[2 marks]
Answer
(c)
Given that $r > \frac{1}{2}$ , find the least $n$ for which $S_{\infty} - S_{n} < 0.5$ .
[4 marks]
Answer
Total for D1: 9 marks

D2
The series $5 + 5 (2 x - 3) + 5 (2 x - 3)^{2} + \dots$ has a sum to infinity. Leo writes: "It converges when $2 x - 3 < 1$ , so when $x < 2$ ."
Leo is wrong.
(a)
Find the correct range of values of $x$ .
[3 marks]
Answer
(b)
Given that the sum to infinity is $4$ , find $x$ .
[3 marks]
Answer
(c)
Explain why the sum to infinity cannot be $2$ .
[2 marks]
Total for D2: 8 marks
D3
The sum to infinity of a geometric series is $\frac{4}{3}$ times the sum of its first two terms. The third term is $6$ . Find the two possible values of the sum to infinity.
[7 marks]
Answer
D4
Consider the series $1 + sin^{2} θ + sin^{4} θ + \dots$ .
(a)
Show that, where it exists, the sum to infinity is $sec^{2} θ$ .
[3 marks]
(b)
Explain why there is no sum to infinity when $θ = \frac{π}{2}$ .
[2 marks]
(c)
Find the values of $θ$ in $0 < θ < π$ for which the sum to infinity is $4$ .
[5 marks]
Answer
Total for D4: 10 marks
D5
The $r$ th term of a series is $u_{r} = 3 \times 2^{- r} + 4 \times 3^{- r}$ , for $r = 1, 2, 3, \dots$
(a)
Show that the sum to infinity of the series is $5$ .
[3 marks]
(b)
Show that $S_{\infty} - S_{n} = 3 \times 2^{- n} + 2 \times 3^{- n}$ .
[3 marks]
(c)
Hence find the least value of $n$ for which $S_{n}$ is within $0.001$ of the sum to infinity.
[3 marks]
Answer
Total for D5: 9 marks
D6
Recurring decimals can be written as geometric series.
(a)
Use a geometric series to show that $0. 9 ˙ = 1$ .
[3 marks]
(b)
The digits $p$ and $q$ are whole numbers from $0$ to $9$ . Prove that $0. p ˙ q ˙ = \frac{10 p + q}{99}$ .
[4 marks]
Total for D6: 7 marks

Extension

Stretch

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

E1
A geometric series has sum to infinity $12$ . The series formed by squaring each of its terms has sum to infinity $48$ . Find the first term and the common ratio, and find the sum to infinity of the series formed by cubing each term.
[9 marks]
Hint 1 · What to try
The squared series has first term $a^{2}$ and common ratio $r^{2}$ .
Hint 2 · The first line
Divide $\frac{a ^{2}}{1 - r ^{2}} = 48$ by $\frac{a}{1 - r} = 12$ . Factorise $1 - r^{2}$ .
Hint 3 · The full method
You should get $\frac{a}{1 + r} = 4$ . The cubed series has ratio $r^{3}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can state the condition $∣ r ∣ < 1$ and use it to find a range of values of $x$ .
I can use $S_{\infty} = \frac{a}{1 - r}$ forwards and backwards, including a negative ratio.
I can write a recurring decimal as a fraction using a geometric series.
I can find the least $n$ for which $S_{n}$ is within a given distance of $S_{\infty}$ .

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