A geometric series has a = 4 and r = 12. Find the sum to infinity. [2 marks]

CCEA CCEA 2020-style practice: A geometric series has first term 8 and common ratio (3)/(4). Find the sum of the first 5 terms and the sum to infinity.

The sum of the first n terms is S_n = (a(1 - r^n))/(1 - r). With a = 8, r = (3)/(4), n = 5: S_5 = (8(1 - (0.75)^5))/(1 - 0.75) = (8(1 - 0.2373))/(0.25) = (8(0.7627))/(0.25) = 24.4. The sum to infinity (valid since |r| < 1) is S_∞ = (a)/(1 - r) = (8)/(0.25) = 32. Markers reward the sum formula, the value of S_5, checking |r| < 1, and the sum to infinity.

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How do arithmetic and geometric series sum, and how does the binomial expansion extend to any power?

Arithmetic and geometric sequences and series and their sums, the sum to infinity of a convergent geometric series, and the binomial expansion for any rational power with its validity condition.

A CCEA A-Level Mathematics answer on arithmetic and geometric sequences and series, the formulae for their sums, the sum to infinity of a convergent geometric series, and the binomial expansion for any rational power with its range of validity.

Generated by Claude Opus 4.813 min answerUpdated 2026-06-16

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
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Try this

What this dot point is asking

CCEA wants you to work with arithmetic and geometric sequences and series, use the formulae for their $n$ th term and their sum, find the sum to infinity of a convergent geometric series, and apply the binomial expansion to any rational power, stating the range of values for which it is valid. This extends the positive-integer binomial of AS 1 and powers many approximations.

The answer

Arithmetic sequences and series

Geometric sequences and series

The general binomial expansion

Unlike the AS expansion for a positive integer power, which terminates after $n + 1$ terms, the expansion for a fractional or negative power is an infinite series that only approximates the function, and only within a restricted range. Exam questions typically ask for the first three or four terms and the values of $x$ for which the expansion is valid.

Worked example: a sum to infinity in context

Find a total distance from a geometric series

A ball is dropped from $2\,\text{m}$ and each bounce reaches $60\%$ of the previous height. Find the total vertical distance travelled before it comes to rest.

step Identify the geometric structure

The ball falls $2\,\text{m}$ , then rises and falls $1.2\,\text{m}$ , then $0.72\,\text{m}$ , and so on. After the first drop, each up-and-down pair is twice a geometric series with $a = 1.2$ and $r = 0.6$ .

step Sum the bouncing part

S_\infty = \frac{a}{1 - r} = \frac{1.2}{1 - 0.6} = \frac{1.2}{0.4} = 3\,\text{m up}.

The ball travels this distance up and the same distance down, so

2 \times 3 = 6\,\text{m}

for the bounces.

step Add the first drop

Total $= 2 + 6 = 8\,\text{m}$ . The sum to infinity captures the endless but finite total of the shrinking bounces.

Examples in context

Example 1. Loan repayments. A reducing-balance loan repaid by equal instalments forms a geometric series, and the sum formula gives the total or the outstanding balance. Geometric series are the mathematics behind annuities and mortgages.

Example 2. Approximating a root. Expanding $(1 + x)^{1/2} \approx 1 + \tfrac{1}{2}x - \tfrac{1}{8}x^2$ gives $\sqrt{1.04} \approx 1 + 0.02 - 0.0002 = 1.0198$ . The general binomial provides quick, controlled approximations to roots and reciprocals.

Try this

Q1. An arithmetic sequence has $a = 5$ and $d = 3$ . Find the 10th term. [2 marks]

Cue. $u_{10} = 5 + 9(3) = 32$ .

Q2. A geometric series has $a = 4$ and $r = \tfrac{1}{2}$ . Find the sum to infinity. [2 marks]

Cue. $S_\infty = \frac{4}{1 - 0.5} = 8$ .

Q3. State the range of validity for the expansion of $(1 + x)^{-2}$ . [1 mark]

Cue. $|x| < 1$ .

Exam-style practice questions

Practice questions written in the style of CCEA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

CCEA 20206 marksA geometric series has first term

8

and common ratio

\frac{3}{4}

. Find the sum of the first

5

terms and the sum to infinity.

Show worked answer →

The sum of the first $n$ terms is $S_n = \frac{a(1 - r^n)}{1 - r}$ .

With $a = 8$ , $r = \frac{3}{4}$ , $n = 5$ :

$S_5 = \frac{8\left(1 - (0.75)^5\right)}{1 - 0.75} = \frac{8(1 - 0.2373)}{0.25} = \frac{8(0.7627)}{0.25} = 24.4.$

The sum to infinity (valid since $|r| < 1$ ) is $S_\infty = \frac{a}{1 - r} = \frac{8}{0.25} = 32$ .

Markers reward the sum formula, the value of $S_5$ , checking $|r| < 1$ , and the sum to infinity.

CCEA 20196 marksFind the first three terms in the binomial expansion of

(1 + 4x)^{1/2}

in ascending powers of

x

, and state the range of values of

x

for which it is valid.

Show worked answer →

The general binomial expansion is $(1 + y)^n = 1 + ny + \frac{n(n - 1)}{2!}y^2 + \dots$ with $y = 4x$ and $n = \frac{1}{2}$ .

First term: $1$ .

Second: $\frac{1}{2}(4x) = 2x$ .

Third: $\frac{\frac{1}{2}\left(\frac{1}{2} - 1\right)}{2}(4x)^2 = \frac{\frac{1}{2}\left(-\frac{1}{2}\right)}{2}(16x^2) = \frac{-\frac{1}{4}}{2}(16x^2) = -2x^2.$

So $(1 + 4x)^{1/2} \approx 1 + 2x - 2x^2$ .

It is valid for $|4x| < 1$ , that is $|x| < \frac{1}{4}$ .

Markers reward the general expansion, each of the first three terms, and the validity condition.

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Sources & how we know this

CCEA GCE Mathematics specification — CCEA (2018)