Arithmetic and Geometric Sequences

Arithmetic and Geometric Sequences

Sequences and series form a major topic at A-Level. You must understand the formulae, prove results using sigma notation, and apply them to real-world problems.

Arithmetic Sequences

An arithmetic sequence has a common difference (d) between consecutive terms. If the first term is a, the sequence is:

a, a + d, a + 2d, a + 3d, ...

The nth term (general term) is:

uₙ = a + (n - 1)d

Worked Example: In an arithmetic sequence, u₃ = 11 and u₇ = 23. Find a and d.

u₃ = a + 2d = 11 ... (i)

u₇ = a + 6d = 23 ... (ii)

Subtracting (i) from (ii): 4d = 12, so d = 3.

From (i): a = 11 - 6 = 5.

Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic series is:

Sₙ = n/2 × (2a + (n - 1)d)

Alternatively, if you know the last term l:

Sₙ = n/2 × (a + l)

This formula arises from pairing terms from opposite ends: u₁ + uₙ = u₂ + uₙ₋₁ = ... = a + l. There are n/2 such pairs.

Worked Example: Find the sum of the first 20 terms of 5, 8, 11, 14, ...

a = 5, d = 3, n = 20.

S₂₀ = 20/2 × (2(5) + 19(3)) = 10 × (10 + 57) = 10 × 67 = 670

Geometric Sequences

A geometric sequence has a common ratio (r) between consecutive terms:

a, ar, ar², ar³, ...

The nth term is:

uₙ = arⁿ⁻¹

To find r, divide any term by the previous term: r = uₙ₊₁/uₙ.

Worked Example: The third term of a geometric sequence is 12 and the sixth term is 96. Find a and r.

u₃ = ar² = 12 ... (i)

u₆ = ar⁵ = 96 ... (ii)

Dividing (ii) by (i): r³ = 8, so r = 2.

From (i): a(4) = 12, so a = 3.

Sum of a Geometric Series

The sum of the first n terms of a geometric series is:

Sₙ = a(1 - rⁿ) / (1 - r) when r ≠ 1

Equivalently: Sₙ = a(rⁿ - 1) / (r - 1)

Use whichever avoids negative values in numerator and denominator.

Worked Example: Find the sum of the first 8 terms of 3, 6, 12, 24, ...

a = 3, r = 2, n = 8.

S₈ = 3(2⁸ - 1)/(2 - 1) = 3(256 - 1)/1 = 3 × 255 = 765

Sum to Infinity

If |r| < 1 (the common ratio is between -1 and 1), the terms get progressively smaller and the series converges. The sum to infinity is:

S∞ = a / (1 - r)

If |r| ≥ 1, the series diverges and has no finite sum.

Worked Example: Find the sum to infinity of 10, -5, 2.5, -1.25, ...

a = 10, r = -0.5. Since |r| = 0.5 < 1, the series converges.

S∞ = 10 / (1 - (-0.5)) = 10/1.5 = 20/3 ≈ 6.667

Sigma Notation

Sigma notation is a compact way to write sums:

Σ from r=1 to n of f(r) means f(1) + f(2) + f(3) + ... + f(n)

Standard results you should know:

• Σ(r=1 to n) 1 = n
• Σ(r=1 to n) r = n(n+1)/2
• Σ(r=1 to n) r² = n(n+1)(2n+1)/6

These can be used to evaluate sums of polynomial expressions term by term.

Worked Example: Evaluate Σ(r=1 to 20) (3r + 1).

= 3·Σr + Σ1 = 3·(20×21/2) + 20 = 3(210) + 20 = 630 + 20 = 650

Recurrence Relations

A recurrence relation defines each term using the previous term(s):

uₙ₊₁ = f(uₙ)

• Arithmetic: uₙ₊₁ = uₙ + d (add a constant)
• Geometric: uₙ₊₁ = r·uₙ (multiply by a constant)
• Other: uₙ₊₁ = uₙ² + 3, for example (neither arithmetic nor geometric)

To determine the behaviour, compute several terms and look for convergence, divergence, or periodicity.

Exam Tips

• Read carefully whether the question asks for the nth term or the sum of n terms — different formulae.
• For sum to infinity, always verify that |r| < 1 and state this condition.
• When given information about two terms, set up simultaneous equations — dividing eliminates a.
• In context (modelling) questions, identify whether the model is arithmetic (linear growth/decay) or geometric (percentage growth/decay).
• Sigma notation questions often require splitting the sum into standard components and evaluating each separately.