Sequences and Series
Sequences and Series tests recognition of AP, GP and HP patterns, standard summation formulas, telescoping, and inequalities involving means.
Core rules
AP: tₙ = a + (n − 1)d and Sₙ = (n/2)(2a + (n − 1)d). GP: tₙ = ar^(n−1), Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, and S∞ = a/(1 − r) only when |r| < 1. For positive numbers AM ≥ GM ≥ HM, GM² = AM × HM, and equality holds only when all the numbers are equal. Standard sums: Σn = n(n + 1)/2, Σn² = n(n + 1)(2n + 1)/6, Σn³ = [n(n + 1)/2]².
Techniques
A series of the form Σ 1/(aₙa_(n+1)) is split by partial fractions into a difference and then telescoped. An arithmetico-geometric series is summed by multiplying by the common ratio and subtracting.
Worked example
Sum 1/(1·3) + 1/(3·5) + … up to n terms. The rth term is 1/((2r − 1)(2r + 1)) = ½[1/(2r − 1) − 1/(2r + 1)]. Summing, every intermediate term cancels, leaving ½[1 − 1/(2n + 1)] = n/(2n + 1).
Traps
Using S∞ when |r| ≥ 1; miscounting the number of terms; applying AM = GM to unequal numbers; dropping the ½ produced by partial fractions.