What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence has a constant difference d between consecutive terms: a, a+d, a+2d, .... The nth term is a + (n-1)d. A geometric sequence has a constant ratio r between consecutive terms: a, ar, ar^2, .... The nth term is ar^(n-1). At A-Level, you need to identify which type applies and use the correct formula.

How do I find the sum of an arithmetic series?

The sum of the first n terms of an arithmetic series is Sn = n/2 * (2a + (n-1)d), which can also be written as Sn = n/2 * (first term + last term). For example, 1+2+3+...+100 has a=1, d=1, n=100: S100 = 100/2 * (2+99) = 50 * 101 = 5050. This is the formula Gauss famously derived as a child.

When does a geometric series converge and what is the sum to infinity?

A geometric series converges (has a finite sum to infinity) when |r| = 1, the series diverges and has no sum to infinity.

What is the Fibonacci sequence and how does it relate to A-Level maths?

The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, ... where each term is the sum of the previous two: F(n) = F(n-1) + F(n-2). At A-Level, it appears in proof questions and as an example of a recurrence relation. The ratio of consecutive Fibonacci terms converges to the golden ratio phi = (1+sqrt(5))/2 ≈ 1.618.

How do I find n given the sum or nth term of a sequence?

For arithmetic sequences: if given the nth term un = a+(n-1)d, rearrange to n = (un - a)/d + 1. If given the sum Sn, use the quadratic formula on n/2*(2a+(n-1)d) = Sn. For geometric sequences: if given un = ar^(n-1), take logarithms to find n = 1 + log(un/a)/log(r). Always check n is a positive integer.

What is sigma notation and how do I use it for series at A-Level?

Sigma notation (capital Greek letter Sigma, written as S) represents a sum. sum[r=1 to n] f(r) means add up f(r) for r = 1, 2, ..., n. For example, sum[r=1 to n] r = n(n+1)/2 (sum of first n natural numbers). At A-Level, you need to recognise and evaluate sums of r, r^2, and r^3 using standard formulae.

How do recurrence relations work in A-Level sequences?

A recurrence relation defines each term from previous terms. For example, u(n+1) = 2*u(n) + 3 with u1 = 1 gives: u2 = 5, u3 = 13, u4 = 29. You may be asked to prove periodicity (does the sequence repeat?), find a specific term, or identify whether it is arithmetic/geometric. Recurrence relations appear in both AS and A2 papers.

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Arithmetic Sequence & Series

An arithmetic sequence has a common difference d between consecutive terms. Enter the first term (a), common difference (d), and number of terms (n).

First term (a)

Common difference (d)

Number of terms (n)

Arithmetic Sequences: Definition and Formulae

An arithmetic sequence (also called an arithmetic progression, or AP) is a sequence in which each term differs from the previous term by a fixed amount called the common difference d. The general form is:

a, a+d, a+2d, a+3d, …, a+(n-1)d

The key formulae for arithmetic sequences at A-Level are:

nth term: u_n = a + (n-1)d
Sum to n terms: S_n = n/2 × (2a + (n-1)d) = n/2 × (first term + last term)
If the last term l = a + (n-1)d is known: S_n = n/2 × (a + l)

Example: Find the sum of the AP: 5, 9, 13, ..., up to 20 terms.
a = 5, d = 4, n = 20
S_20 = 20/2 × (2×5 + 19×4) = 10 × (10 + 76) = 10 × 86 = 860

Geometric Sequences: Definition and Formulae

A geometric sequence (geometric progression, GP) has a constant ratio r between successive terms. If the first term is a, the sequence is:

a, ar, ar^2, ar^3, …, ar^(n-1)

Key formulae at A-Level:

nth term: u_n = ar^(n-1)
Sum to n terms: S_n = a(1 - r^n) / (1 - r) for r ≠ 1
Sum to infinity (when |r| < 1): S_∞ = a / (1 - r)

Example: Find S_∞ for the GP: 12, 6, 3, 1.5, ...
a = 12, r = 0.5, |r| = 0.5 < 1 ✓
S_∞ = 12 / (1 - 0.5) = 12 / 0.5 = 24

Fibonacci Sequence and the Golden Ratio

The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... and is defined by the recurrence relation F(n) = F(n-1) + F(n-2) with F(1) = F(2) = 1. It appears in nature (phyllotaxis, shell spirals) and in computer science. The ratio of consecutive Fibonacci terms converges to the golden ratio φ = (1 + √5) / 2 ≈ 1.6180339...

At A-Level, Fibonacci-type sequences appear in recurrence relation questions. You may be asked to prove that a recurrence relation generates the sequence or to find the 10th term given the rule u(n+2) = u(n+1) + u(n).

Sigma Notation and Standard Sums

The sigma symbol Σ represents summation. At A-Level, you need these standard results:

Series	Formula
Σr (r=1 to n)	n(n+1)/2
Σr² (r=1 to n)	n(n+1)(2n+1)/6
Σr³ (r=1 to n)	[n(n+1)/2]²
Σ1 (r=1 to n)	n

Convergence and Divergence of Series

A series converges if its sum approaches a finite value as the number of terms increases to infinity. A series diverges if no such finite limit exists. For geometric series:

|r| < 1: Series converges to S_∞ = a/(1-r)
|r| ≥ 1: Series diverges (no sum to infinity)
r = 1: Each term equals a; series diverges to infinity
r = -1: Terms alternate +a, -a; series oscillates and diverges

Arithmetic series always diverge (unless a = d = 0) because the terms grow without bound. Only geometric series with |r| < 1 converge to a finite sum to infinity.

Recurrence Relations at A-Level

A recurrence relation defines each term in terms of previous terms. The general form is u(n+1) = f(u(n)). For example:

u(n+1) = u(n) + d defines an arithmetic sequence (common difference d)
u(n+1) = r × u(n) defines a geometric sequence (common ratio r)
u(n+1) = u(n) + u(n-1) defines the Fibonacci-type sequence
u(n+1) = k - u(n) defines a periodic sequence (period 2)