Series

CIE A-Level Mathematics· Unit 1: Section 6 - Series· 20 min read

1. Arithmetic Progressions★★☆☆☆⏱ 5 min

📘 Definition

Arithmetic Progression (AP)

$a$ = first term, $d$ = common difference, $n$ = number of terms

A sequence where each consecutive term changes by a constant value (which may be positive, negative or zero), called the common difference $d$

Example:

$3, 7, 11, 15...$ has $a = 3$ , $d = 4$

Three-term test: three numbers $a$ , $b$ , $c$ are in arithmetic progression if and only if $2 b = a + c$ (the middle term is the mean of its two neighbours).

For any AP, the nth term is given by $u_{n} = a + (n - 1) d$ . The sum of the first $n$ terms can be calculated using either form of the identity:

S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)

Where $l$ is the last term of the finite AP.

📐 Worked Example

The 3rd term of an AP is 10, and the 7th term is 22. Find the first term, common difference, and sum of the first 10 terms.

1
Set up equations from the nth term formula:
2
$u_3 = a + 2d = 10 \\ u_7 = a + 6d = 22$
3
Subtract the first equation from the second to eliminate $a$ :
4
$4d = 12 \implies d = 3$
5
Substitute $d = 3$ back to find $a$ :
6
$a + 2(3) = 10 \implies a = 4$
7
Calculate the sum of the first 10 terms:
8
$S_{10} = \frac{10}{2}[2(4) + 9(3)] = 5(8 + 27) = 175$

2. Geometric Progressions & Sum to Infinity★★★☆☆⏱ 7 min

📘 Definition

Geometric Progression (GP)

$a$ = first term, $r$ = common ratio

A sequence where each consecutive term is multiplied by a constant value, called the common ratio $r$

Example:

$2, 6, 18, 54...$ has $a = 2$ , $r = 3$

Three-term test: three numbers $a$ , $b$ , $c$ are in geometric progression if and only if $b^{2} = a c$ (the middle term is the geometric mean of its two neighbours). A single question may involve more than one progression, so identify each sequence carefully.

For any GP, the nth term is $u_{n} = a r^{n - 1}$ . The sum of the first $n$ terms is:

S_n = \frac{a(1-r^n)}{1-r}, \quad r \neq 1

📐 Worked Example

A GP has first term 8 and sum to infinity 32. Find the common ratio and the 5th term.

1
Use the sum to infinity formula to solve for $r$ :
2
$S_\infty = \frac{a}{1-r} \implies 32 = \frac{8}{1-r}$
3
Rearrange and solve:
4
$32(1-r) = 8 \implies 1-r = \frac{1}{4} \implies r = \frac{3}{4}$
5
Calculate the 5th term using the nth term formula:
6
$u_5 = ar^4 = 8 \times \left(\frac{3}{4}\right)^4 = 8 \times \frac{81}{256} = \frac{81}{32}$

3. Binomial Expansion for Positive Integer n★★★☆☆⏱ 8 min

📘 Definition

Binomial Theorem

$(r n)$ = binomial coefficient, $n!$ = n factorial

For any positive integer $n$ , the expansion of $(a + b)^{n}$ is a finite series of terms with coefficients given by binomial coefficients

Example:

$(x + 2)^{2} = x^{2} + 4 x + 4$

The full expansion is written as:

(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r

Where the binomial coefficient is calculated as:

\binom{n}{r} = \frac{n!}{r!(n-r)!} = \frac{n(n-1)(n-2)...(n-r+1)}{r!}

📐 Worked Example

Find the first three terms in descending powers of $x$ of $(2 x - \frac{1}{x})^{4}$ .

1
Identify $a = 2 x$ , $b = - \frac{1}{x}$ , $n = 4$ . The first three terms correspond to $k = 0$ , $k = 1$ , $k = 2$ :
2
For $k = 0$ :
3
$\binom{4}{0}(2x)^4 \left(-\frac{1}{x}\right)^0 = 1 \times 16x^4 \times 1 = 16x^4$
4
For $k = 1$ :
5
$\binom{4}{1}(2x)^3 \left(-\frac{1}{x}\right)^1 = 4 \times 8x^3 \times \left(-\frac{1}{x}\right) = -32x^2$
6
For $k = 2$ :
7
$\binom{4}{2}(2x)^2 \left(-\frac{1}{x}\right)^2 = 6 \times 4x^2 \times \frac{1}{x^2} = 24$
8
The first three terms are $16 x^{4} - 32 x^{2} + 24$ .

📐 Worked Example

In the expansion of $(1 - \frac{x}{3})^{n}$ (where $n$ is a positive integer), the coefficient of $x^{2}$ is 4. Find the value of $n$ .

1
The term in $x^{2}$ is $(2 n) (1)^{n - 2} (- \frac{x}{3})^{2}$ , so its coefficient is:
2
$\binom{n}{2}\left(-\frac{1}{3}\right)^2 = \binom{n}{2} \times \frac{1}{9}$
3
Write $(2 n) = \frac{n ( n - 1 )}{2}$ , so the coefficient of $x^{2}$ becomes:
4
$\frac{n(n-1)}{2} \times \frac{1}{9} = \frac{n(n-1)}{18}$
5
Set this equal to 4 and solve for $n$ :
6
$\frac{n(n-1)}{18} = 4 \implies n(n-1) = 72 \implies n^2 - n - 72 = 0$
7
Factorise and take the positive integer root ( $n$ must be a positive integer):
8
$(n-9)(n+8) = 0 \implies n = 9$

4. Common Pitfalls

Wrong move:

Using $n$ instead of $(n - 1)$ in the nth term formula for AP/GP

Why:

The first term corresponds to $n = 1$ , so the offset is $(n - 1)$ , not $n$ .

Correct move:

Always write $u_{n} = a + (n - 1) d$ for AP and $u_{n} = a r^{n - 1}$ for GP, and check with $n = 1$ to confirm.

Wrong move:

Calculating sum to infinity when $∣ r ∣ \geq 1$

Why:

Students often forget that convergence only occurs when $∣ r ∣ < 1$ , and confuse finite sum with infinite sum.

Correct move:

Always check the value of $∣ r ∣$ first: if $∣ r ∣ \geq 1$ , state that sum to infinity does not exist.

Wrong move:

Forgetting negative signs when expanding $(a - b)^{n}$

Why:

When the second term of the binomial is negative, alternate terms will have negative signs, which are often omitted.

Correct move:

Always substitute the full negative value of $b$ into the expansion, and check the sign of each term explicitly.

Wrong move:

Thinking the finite GP sum formula $\frac{a ( 1 - r ^{n} )}{1 - r}$ only works for $r < 1$

Why:

The formula works for all $r \neq = 1$ , only sum to infinity requires $∣ r ∣ < 1$ .

Correct move:

Use the same formula for any $r \neq = 1$ , rearrange to $\frac{a ( r ^{n} - 1 )}{r - 1}$ if you prefer to avoid negative values.

5. Quick Reference Cheatsheet

Series Type	nth Term	Sum Formula	Key Condition
Arithmetic Progression	$a + (n - 1) d$	$\frac{n}{2} (a + l)$	None
Finite Geometric Progression	$a r^{n - 1}$	$\frac{a ( 1 - r ^{n} )}{1 - r} (r \neq = 1)$	None
Infinite Geometric Progression		$\frac{a}{1 - r}$	$∣ r ∣ < 1$ (convergent)
Binomial Expansion (n ∈ ℕ⁺)	$(r n) a^{n - r} b^{r}$	$\sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$	$(r n) = \frac{n !}{r ! ( n - r )!}$

Going deeper

syllabusCIE 9709 Pure Mathematics 1 Syllabus

What's Next

Series is a core topic that appears frequently in Paper 1, often combined with other topics like coordinate geometry, logarithms, and functions. Mastery of binomial expansions is required for later work on negative and fractional binomial expansions in Pure Mathematics 3, and convergence of geometric series lays the foundation for infinite Maclaurin series in A2-level work. You can deepen your understanding of binomial expansions with focused practice, then move on to the next core topic of differentiation.

→Differentiation
→Integration