A-Level — mathsStandard1 min read

Arithmetic and Geometric Sequences and Series

Work with arithmetic and geometric sequences at A-Level. Find nth terms, sums, and sum to infinity.

Tutor AI Team2026-02-25T01:57:10.944Z

Arithmetic and geometric sequences are fundamental to A-Level Maths, with applications in financial maths, modelling, and proofs.

Arithmetic Sequences

Common difference $d$ . $u_n = a + (n-1)d$ .

Sum: $S_n = \frac{n}{2}(2a + (n-1)d) = \frac{n}{2}(a + l)$ where $l$ = last term.

Common ratio $r$ . $u_n = ar^{n-1}$ .

Sum: $S_n = \frac{a(1-r^n)}{1-r}$ for $r \neq 1$ .

For $|r| < 1$ : $S_\infty = \frac{a}{1-r}$ .

Problem

$a = 3$ , $d = 4$ . Find $u_{20}$ and $S_{20}$ .

$u_{20} = 3 + 19(4) = 79$ . $S_{20} = \frac{20}{2}(3 + 79) = 820$ .

Solution

Problem

$a = 2$ , $r = \frac{1}{3}$ . $S_\infty = \frac{2}{1-\frac{1}{3}} = 3$ .

Solution

Problem

The 3rd term of GP is 12 and 6th term is $\frac{3}{2}$ . Find $a$ and $r$ .

$ar^2 = 12$ , $ar^5 = \frac{3}{2}$ . Divide: $r^3 = \frac{1}{8}$ → $r = \frac{1}{2}$ . $a = 48$ .

Solution

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