Finite geometric sums | Year 11

01 / LEARN

The idea, step by step

Subtracting a shifted sum

Write the sum $S_{n} = a + a r + \dots + a r^{n - 1}$ , then multiply it by $r$ . Subtracting cancels the middle terms and gives $(1 - r) S_{n} = a (1 - r^{n})$ . Divide only when $r \neq 1$ .

The equivalent form $a (r^{n} - 1) / (r - 1)$ is convenient when $r > 1$ . Finite sums do not require the convergence condition used for infinite series.

S_{n} = \frac{a (1 - r^{n})}{1 - r} (r \neq 1); S_{n} = n a (r = 1)

Identify the actual powers

In an alternating series, retain brackets around a negative ratio before raising it to a power. Count terms rather than guessing the exponent from the last visible term.

For equal deposits at the end of each period, the newest deposit has earned no interest when observed immediately after payment. Its contribution has exponent zero.

A common trap

The fraction formula applies when $r = 1$ .

Instead: Its denominator would be zero; add $n$ equal terms to get $n a$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

An alternating finite sum

Find the sum of the first six terms of $3, - 6, 12, \dots$ .

$a = 3, r = - 2, n = 6$ .
$S_{6} = 3 [1 - {(- 2)}^{6}] / 3 = - 63$ .

- 63

Example 2

End-of-year deposits

AUD 200 is paid at the end of each year for four years, with 5% annual interest. Find the balance immediately after payment 4.

Contributions are $200 {(1.05)}^{3}$ , $200 {(1.05)}^{2}$ , $200 (1.05)$ and $200$ .
Total $200 [{(1.05)}^{4} - 1] / 0.05 = 862.025$ .

862.03

to the nearest cent.

Check your understanding

Find $1 + 3 + 9 + 27 + 81$ .

Show answer

$121$ .

$(3^{5} - 1) / (3 - 1) = 121$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the sum of eight terms of $5, 10, 20, \dots$ .
2 marks
Find $16 + 8 + 4 + 2 + 1 + 1 / 2$ .
2 marks
Find the seven-term sum when $a = 4, r = 1$ .
2 marks
Find $2 - 6 + 18 - 54 + 162$ .
2 marks
For $a = 7, r = 2$ , find the least $n$ with $S_{n} > 1000$ .
2 marks
Find the balance immediately after three end-of-year AUD 500 deposits earning 4% yearly.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the sum of eight terms of $5, 10, 20, \dots$ .
2 marks
$5 (2^{8} - 1) = 1275$ .
$1275$ .
Find $16 + 8 + 4 + 2 + 1 + 1 / 2$ .
2 marks
$16 (1 - 2^{- 6}) / (1 - 1 / 2) = 63 / 2$ .
$63 / 2$ .
Find the seven-term sum when $a = 4, r = 1$ .
2 marks
Seven equal terms total $7 (4) = 28$ .
$28$ .
Find $2 - 6 + 18 - 54 + 162$ .
2 marks
$2 [1 - {(- 3)}^{5}] / 4 = 122$ .
$122$ .
For $a = 7, r = 2$ , find the least $n$ with $S_{n} > 1000$ .
2 marks
$S_{7} = 889$ and $S_{8} = 1785$ .
$8$ .
Find the balance immediately after three end-of-year AUD 500 deposits earning 4% yearly.
3 marks
$500 [{(1.04)}^{2} + 1.04 + 1] = 1560.8$ .
$1560.80$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics sequence and series content with introductory induction, regrouped into an original school-sized chapter.

Official curriculum source ↗

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