Geometric sequences | Year 11

01 / LEARN

Understand the idea.

Constant multiplication

A geometric sequence multiplies by a fixed ratio at each step. For nonzero consecutive terms, calculate $r$ as later term divided by earlier term. With $u_{1} = a$ , term $n$ requires $n - 1$ multiplications, so $u_{n} = a r^{n - 1}$ .

For a positive first term, $0 < r < 1$ gives positive decay and $r > 1$ gives growth. A negative ratio alternates signs. Track magnitude and sign separately: a negative ratio does not mean every term is negative.

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

More than one possible ratio

Dividing two nonzero known terms gives $u_{q} / u_{p} = r^{q - p}$ . If the index gap is even, a positive quotient may allow both a positive and a negative real ratio. Check both against every supplied condition.

The exponent depends on the starting index. A model beginning at $u_{0}$ uses $u_{n} = u_{0} r^{n}$ . For threshold questions with positive ratios, logarithms can locate a boundary, but an actual term still requires an integer index and a neighbouring-term check.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Alternating geometric terms

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

Worked solution

1
Check the common ratio.
$\frac{- 18}{6} = - 3, \frac{54}{- 18} = - 3$
Both successive quotients agree.
2
Count the multiplications to term seven.
$u_{7} = 6 {(- 3)}^{7 - 1} = 6 {(- 3)}^{6}$
The first term corresponds to exponent zero.
3
Evaluate the signed power.
${(- 3)}^{6} = 729, u_{7} = 6 (729) = 4374$
An even power of a negative ratio is positive.

Answer

$4374$ .

EXAMPLE 02

Two possible ratios

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

Worked solution

1
Use the first and third terms.
$u_{3} = 3 r^{2} = 12$
Two multiplications separate these indices.
2
Retain both real ratios.
$r^{2} = 4 ⟹ r = 2 or r = - 2$
An even power does not determine the sign of its real base.
3
Calculate each possible second term.
$u_{2} = 3 r = 6 or - 6$
Both ratios return the third term to twelve.

Answer

$u_{2} = 6$ or $- 6$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$5$ .

Term five is four halvings after the first: $u_{5} = 80 {(1 / 2)}^{4} = 80 / 16 = 5$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Find $r, u_{6}$ for $4, 12, 36, \dots$ .
2 marks
Find $u_{8}$ if $a = 128, r = - \frac{1}{2}$ .
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 $a = 128, r = - \frac{1}{2}$ .
2 marks
$128 {(- \frac{1}{2})}^{7} = - 1$ .
$- 1$ .
A positive geometric sequence has $u_{2} = 10, u_{5} = 80$ . Find $a, r$ .
2 marks
$r^{3} = \frac{80}{10} = 8$ and $a = \frac{10}{2} = 5$ .
$a = 5, r = 2$ .
Is $2, 6, 18, 55$ geometric?
2 marks
The first two ratios are 3, but $\frac{55}{18} \neq 3$ .
No.
Find both real geometric sequences with $u_{1} = 8, u_{3} = 2$ . Give the first four terms.
2 marks
$r^{2} = \frac{1}{4}$ , so $r = \pm \frac{1}{2}$ .
$8, 4, 2, 1$ 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 ↗

In this lesson

Understand the idea.

Constant multiplication

More than one possible ratio

See the method in action.

Alternating geometric terms

Check the common ratio.

Count the multiplications to term seven.

Evaluate the signed power.

Two possible ratios

Use the first and third terms.

Retain both real ratios.

Calculate each possible second term.

Try it yourself.

Check your understanding

Keep practising