Sequences and Series
Sequences and Series tests arithmetic, geometric and harmonic progressions, the three means and their inequality, and the standard summation formulae. Numerical questions are common, but so are short proofs about means.
Progressions
An AP has nth term a + (n − 1)d and sum (n/2)(2a + (n − 1)d). A GP has nth term arⁿ⁻¹ and sum a(rⁿ − 1)/(r − 1) for r ≠ 1; the sum to infinity a/(1 − r) is valid only when |r| < 1. A harmonic progression is the reciprocal of an AP, so the harmonic mean of a and b is 2ab/(a + b), the reciprocal of the arithmetic mean of 1/a and 1/b.
Means and standard sums
For positive numbers AM ≥ GM ≥ HM, with equality only when the numbers are equal; also AM × HM = GM². The standard sums are Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6 and Σn³ = [n(n+1)/2]².
Worked example
An infinite GP has first term 2 and sum 8. Using a/(1 − r) = 8 gives 2/(1 − r) = 8, so 1 − r = 1/4 and r = 3/4. Because |3/4| < 1, the infinite sum genuinely converges to 8.
Traps
- Using a/(1 − r) when |r| ≥ 1, where the series diverges.
- Confusing the number of terms with the index of the nth term.
- Applying AM ≥ GM to a set containing negative numbers.
- Inverting the harmonic mean formula carelessly.