Sequences and Series

Sequences and series

A sequence is an ordered list of terms; a series is the sum of the terms. Sigma notation (Σ) is used for sums.

Arithmetic sequences

Each term increases by a constant common difference d.

• nth term: uₙ = a + (n − 1)d (a = first term).
• Sum of n terms:

Sₙ = (n/2)[2a + (n − 1)d]   or   Sₙ = (n/2)(a + l)

(l = last term).

Geometric sequences

Each term is multiplied by a constant common ratio r.

• nth term: uₙ = ar^(n−1).
• Sum of n terms:

Sₙ = a(1 − rⁿ) ÷ (1 − r)

• Sum to infinity (only if |r| < 1, so the terms shrink):

S∞ = a ÷ (1 − r)

Recurrence relations

A sequence can be defined by a rule linking one term to the next, e.g. u_(n+1) = 2uₙ + 1, with a starting value. Some sequences are increasing, decreasing or periodic.

Worked example

A geometric series has first term 8 and common ratio 0.5. Find its sum to infinity.

• |r| = 0.5 < 1, so S∞ exists.
• S∞ = a ÷ (1 − r) = 8 ÷ (1 − 0.5) = 8 ÷ 0.5 = 16. ✓

Common mistakes

• Using the arithmetic formulas for a geometric sequence (or vice versa).
• Applying S∞ when |r| ≥ 1 (it only converges for |r| < 1).
• Off-by-one errors with (n − 1) in the nth-term formulas.

Exam tips

• Identify arithmetic (add d) vs geometric (multiply by r) first.
• Check the |r| < 1 condition before using sum to infinity.
• Learn both forms of the arithmetic sum (with last term l, or with d).

Key facts to remember

• Arithmetic: uₙ = a + (n−1)d; Sₙ = (n/2)[2a + (n−1)d].
• Geometric: uₙ = ar^(n−1); Sₙ = a(1−rⁿ)/(1−r); S∞ = a/(1−r) only if |r| < 1.
• Sequences may be defined by recurrence relations and be increasing/decreasing/periodic.