Geometric sequences | Year 11

01 / LEARN

The idea, step by step

Constant multiplication

A geometric sequence multiplies each term by a fixed ratio $r$ . For nonzero consecutive terms, divide the later term by the earlier term. With first term $u_{1} = a$ , the first exponent is zero.

A negative ratio alternates the signs. Positive ratios smaller than 1 produce decay when the first term is positive. A decreasing list need not be geometric.

u_{n} = a r^{n - 1}

More than one possible ratio

Non-adjacent terms give a power of $r$ . An even power may permit two real ratios, one positive and one negative. Retain both unless the context or another supplied term rules one out.

A threshold calculation may use logarithms, but the answer must still be an allowed integer index. Check the adjacent terms around the threshold.

A common trap

With first index 1 the formula is $a r^{n}$ .

Instead: Use $a r^{n - 1}$ ; an index-zero model instead uses $u_{0} r^{n}$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Alternating geometric terms

For $6, - 18, 54, \dots$ , find $u_{7}$ .

$r = - 3$ .
$u_{7} = 6 {(- 3)}^{6} = 4374$ .

4374

Example 2

Two possible ratios

$u_{1} = 3, u_{3} = 12$ in a real geometric sequence. Find all possible second terms.

$3 r^{2} = 12$ gives $r = \pm 2$ .
Then $u_{2} = 3 r = \pm 6$ .

u_{2} = 6

- 6

Check your understanding

For $a = 80, r = 1 / 2$ , find $u_{5}$ .

Show answer

$5$ .

$80 {(1 / 2)}^{4} = 5$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Find $r, u_{6}$ for $4, 12, 36, \dots$ .
2 marks
Find $u_{8}$ if $a = 128, r = - 1 / 2$ .
2 marks
A positive geometric sequence has $u_{2} = 10, u_{5} = 80$ . Find $a, r$ .
2 marks
Is $2, 6, 18, 55$ geometric?
2 marks
Find both real geometric sequences with $u_{1} = 8, u_{3} = 2$ . Give the first four terms.
2 marks
A culture has 300 cells at hour zero and doubles hourly. Give its rule and first whole hour above 10,000.
3 marks

Read the worked solutions

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

Find $r, u_{6}$ for $4, 12, 36, \dots$ .
2 marks
$4 (3^{5}) = 972$ .
$r = 3, u_{6} = 972$ .
Find $u_{8}$ if $a = 128, r = - 1 / 2$ .
2 marks
$128 {(- 1 / 2)}^{7} = - 1$ .
$- 1$ .
A positive geometric sequence has $u_{2} = 10, u_{5} = 80$ . Find $a, r$ .
2 marks
$r^{3} = 80 / 10 = 8$ and $a = 10 / 2 = 5$ .
$a = 5, r = 2$ .
Is $2, 6, 18, 55$ geometric?
2 marks
The first two ratios are 3, but $55 / 18 \neq 3$ .
No.
Find both real geometric sequences with $u_{1} = 8, u_{3} = 2$ . Give the first four terms.
2 marks
$r^{2} = 1 / 4$ , so $r = \pm 1 / 2$ .
$8, 4, 2, 1$ and $8, - 4, 2, - 1$ .
A culture has 300 cells at hour zero and doubles hourly. Give its rule and first whole hour above 10,000.
3 marks
$N_{5} = 9600$ and $N_{6} = 19200$ .
$N_{n} = 300 (2^{n})$ ; hour 6.

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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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