Geometric sequences (the general term)

Covers AS91258

What a geometric sequence is

A geometric sequence multiplies by the same number every step.
That fixed multiplier is the common ratio, $r$ — found by dividing any term by the one before it.
Example — $3, 6, 12, 24, \ldots$ has $a = 3$ and $r = 2$ .
If $r$ is between $-1$ and $1$ the terms shrink; if $|r| > 1$ they grow; a negative $r$ makes them alternate in sign.

A bar chart of the sequence 3, 6, 12, 24 where each bar is double the last, showing a common ratio of 2.

The general term

$u_n = a\,r^{\,n-1}$

$a$ — the first term.
$r$ — the common ratio.
$n$ — the position; the power is $n - 1$ (the first term has been multiplied by $r$ zero times).

Finding the common ratio

Divide consecutive terms: $r = \dfrac{u_2}{u_1} = \dfrac{u_3}{u_2}$ .
Check it is constant — if the ratio changes, the sequence is not geometric.

Finding the number of terms (using logarithms)

To find which term reaches a certain size, set $u_n$ equal to that value and solve for $n$ .
Because $n$ is in the power, you take logarithms of both sides to bring it down.

Worked ExampleFinding a term

A population of bacteria triples every hour. If it starts at 500, how many are there after 6 hours?

Step 1 — Identify $a$ and $r$

Start $a = 500$ ; tripling means $r = 3$ . "After 6 hours" is the 7th term (hour 0 is the 1st), or use $500 \times 3^{6}$ directly.

Step 2 — Apply the multiplier six times

$500 \times 3^{6} = 500 \times 729 = 364{,}500$

There are 364,500 bacteria after 6 hours.

Worked ExampleSolving for the number of terms with logs

The geometric sequence $5, 10, 20, \ldots$ — which is the first term to exceed 1000?

Step 1 — Set the general term above 1000

$a = 5$ , $r = 2$ , so $u_n = 5 \times 2^{\,n-1} > 1000 \Rightarrow 2^{\,n-1} > 200$ .

Step 2 — Take logarithms to release $n$

$(n - 1)\log 2 > \log 200 \;\Rightarrow\; n - 1 > \frac{\log 200}{\log 2} = 7.64\ldots$

Step 3 — Round up to a whole term

$n - 1 > 7.64\ldots$ so $n > 8.64\ldots$ , meaning $n = 9$ . The 9th term ( $5 \times 2^{8} = 1280$ ) is the first to exceed 1000.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A geometric sequence has $a = 2$ and $r = 3$ . Find the 5th term.

Merit

The 2nd term of a geometric sequence is 12 and the 4th term is 108. Find the common ratio and the first term (take $r > 0$ ).

Excellence

An investment of $2000 grows by 6% each year, so its value is $2000 \times 1.06^{\,n}$ after $n$ years. In which year does it first double?

Geometric sequences (the general term)

Covers AS91258

What a geometric sequence is

A geometric sequence multiplies by the same number every step.
That fixed multiplier is the common ratio, $r$ — found by dividing any term by the one before it.
Example — $3, 6, 12, 24, \ldots$ has $a = 3$ and $r = 2$ .
If $r$ is between $-1$ and $1$ the terms shrink; if $|r| > 1$ they grow; a negative $r$ makes them alternate in sign.

A bar chart of the sequence 3, 6, 12, 24 where each bar is double the last, showing a common ratio of 2.

The general term

$u_n = a\,r^{\,n-1}$

$a$ — the first term.
$r$ — the common ratio.
$n$ — the position; the power is $n - 1$ (the first term has been multiplied by $r$ zero times).

Finding the common ratio

Divide consecutive terms: $r = \dfrac{u_2}{u_1} = \dfrac{u_3}{u_2}$ .
Check it is constant — if the ratio changes, the sequence is not geometric.

Finding the number of terms (using logarithms)

To find which term reaches a certain size, set $u_n$ equal to that value and solve for $n$ .
Because $n$ is in the power, you take logarithms of both sides to bring it down.

Worked ExampleFinding a term

A population of bacteria triples every hour. If it starts at 500, how many are there after 6 hours?

Step 1 — Identify $a$ and $r$

Start $a = 500$ ; tripling means $r = 3$ . "After 6 hours" is the 7th term (hour 0 is the 1st), or use $500 \times 3^{6}$ directly.

Step 2 — Apply the multiplier six times

$500 \times 3^{6} = 500 \times 729 = 364{,}500$

There are 364,500 bacteria after 6 hours.

Worked ExampleSolving for the number of terms with logs

The geometric sequence $5, 10, 20, \ldots$ — which is the first term to exceed 1000?

Step 1 — Set the general term above 1000

$a = 5$ , $r = 2$ , so $u_n = 5 \times 2^{\,n-1} > 1000 \Rightarrow 2^{\,n-1} > 200$ .

Step 2 — Take logarithms to release $n$

$(n - 1)\log 2 > \log 200 \;\Rightarrow\; n - 1 > \frac{\log 200}{\log 2} = 7.64\ldots$

Step 3 — Round up to a whole term

$n - 1 > 7.64\ldots$ so $n > 8.64\ldots$ , meaning $n = 9$ . The 9th term ( $5 \times 2^{8} = 1280$ ) is the first to exceed 1000.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A geometric sequence has $a = 2$ and $r = 3$ . Find the 5th term.

Merit

The 2nd term of a geometric sequence is 12 and the 4th term is 108. Find the common ratio and the first term (take $r > 0$ ).

Excellence

An investment of $2000 grows by 6% each year, so its value is $2000 \times 1.06^{\,n}$ after $n$ years. In which year does it first double?

Geometric sequences (the general term)

What a geometric sequence is

The general term

Finding the common ratio

Finding the number of terms (using logarithms)

Step 1 — Identify aaa and rrr

Step 2 — Apply the multiplier six times

Step 1 — Set the general term above 1000

Step 2 — Take logarithms to release nnn

Step 3 — Round up to a whole term

Test yourself

Practice by grade

Geometric sequences (the general term)

What a geometric sequence is

The general term

Finding the common ratio

Finding the number of terms (using logarithms)

Step 1 — Identify aaa and rrr

Step 2 — Apply the multiplier six times

Step 1 — Set the general term above 1000

Step 2 — Take logarithms to release nnn

Step 3 — Round up to a whole term

Test yourself

Practice by grade

Step 1 — Identify $a$ and $r$

Step 2 — Take logarithms to release $n$

Step 1 — Identify $a$ and $r$

Step 2 — Take logarithms to release $n$