Sequences and Series
Core ideas
Arithmetic and geometric sequences, their nth terms and sums, sigma notation, sum to infinity of a convergent geometric series, and the binomial expansion for positive integer and (with care) rational indices. Convergence conditions and recurrence relations are also tested.
Traps
- The infinite geometric sum S = a/(1-r) is valid only when |r| < 1; state this condition.
- Confusing the nth term with the sum: the nth term of an AP is a + (n-1)d, but the sum is n/2[2a + (n-1)d].
- In binomial expansion with negative or fractional index, the expansion is valid only for |x| < 1 (or the appropriate scaled condition).
- With recurrence relations, compute several terms to detect the pattern before assuming arithmetic or geometric behaviour.
Rule of thumb
For a geometric series, identify a (first term) and r (common ratio) precisely. The sum to n terms is a(1-r^n)/(1-r); as n grows, r^n tends to zero only if |r| < 1, giving S = a/(1-r).
Worked example
A geometric series has second term 6 and fifth term 48. Then ar = 6 and ar^4 = 48, so r^3 = 8, r = 2, a = 3. The sum to infinity is a/(1-r) = 3/(1-2) = -3 - but |r| = 2 > 1, so the series diverges and no finite sum to infinity exists. This illustrates why the convergence condition must be checked before applying the formula.