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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).

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:

SeriesFormula
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)

How the Sequences &amp; Series Calculator Works

This calculator uses the current UK grading system and educational standards to help students, parents, and teachers understand academic performance. UK education follows specific grading frameworks that differ between GCSEs, A-Levels, and university degrees.

Understanding how grades are calculated and what they mean for future progression is important for making informed decisions about subject choices, university applications, and career planning.

Key Information for 2025/26

GCSEs in England use the 9-1 grading scale, where 9 is the highest and 4 is a standard pass (equivalent to the old C grade). A-Levels use the A*-E scale with UCAS tariff points ranging from 56 (A*) to 16 (E). Universities typically require grades between AAA and CCC depending on the course and institution, with highly competitive courses often asking for A*A*A.

Example Calculation

A student achieving grades of A*, A, and B at A-Level would earn UCAS tariff points of 56 + 48 + 40 = 144 points total. This exceeds the typical entry requirement for most Russell Group universities (128 points or AAB equivalent) and would make the student competitive for many courses.

Source: Based on Ofqual and UCAS 2025/26 guidelines. Last updated March 2026.

Frequently Asked Questions

An arithmetic sequence has a constant difference d: a, a+d, a+2d, ... with nth term a+(n-1)d. A geometric sequence has a constant ratio r: a, ar, ar^2, ... with nth term ar^(n-1). The key difference is addition (AP) versus multiplication (GP).
Sn = n/2 * (2a + (n-1)d). This is also equal to n/2 * (first term + last term). For 1+2+...+100: a=1, d=1, n=100, S=5050.
A geometric series converges when |r| < 1. S_inf = a/(1-r). For 1+0.5+0.25+...: a=1, r=0.5, S_inf = 2. If |r| >= 1, the series diverges.
F(n) = F(n-1) + F(n-2), with F(1) = F(2) = 1. The sequence is 1,1,2,3,5,8,13,21,... At A-Level it is used as an example of a recurrence relation. The ratio F(n+1)/F(n) converges to the golden ratio phi = (1+sqrt(5))/2 ≈ 1.618.
For arithmetic: from nth term un = a+(n-1)d, solve n = (un-a)/d + 1. From sum, use the quadratic formula. For geometric: n = 1 + log(un/a)/log(r). Always verify n is a positive integer.
Sigma notation sums a function over a range of integers. Key results: sum[r=1 to n] r = n(n+1)/2, sum r^2 = n(n+1)(2n+1)/6, sum r^3 = [n(n+1)/2]^2. Use linearity to split and simplify composite sums.
A recurrence relation defines each term from previous ones: u(n+1) = f(u(n)). Common types: u(n+1) = u(n)+d (AP), u(n+1) = r*u(n) (GP), u(n+1) = k-u(n) (periodic, period 2). You may need to find specific terms or prove periodicity.