Sequences and Series Notes — JC H2

nth term from the sum to n terms

Sₙ = T₁ + T₂ + … + Tₙ; T₁ = S₁ and Tₙ = Sₙ − Sₙ₋₁ for n ≥ 2

Applies to all types of sequences and series. Sₙ is the sum of the first n terms and Tₙ is the nth term.

Arithmetic progression (AP)

$T_n = a + (n - 1)d, \quad S_n = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)$

a is the first term, d is the common difference and l = a + (n − 1)d is the last term. A sequence is an AP if Tₙ − Tₙ₋₁ is a constant independent of n. Sum to infinity is not applicable to an AP.

Geometric progression (GP)

$T_n = ar^{n-1}, \quad S_n = \frac{a(1 - r^n)}{1 - r}, \quad S_\infty = \frac{a}{1 - r}$

a is the first term and r is the common ratio. S∞ exists if and only if |r| < 1, since rⁿ → 0 as n → ∞ when |r| < 1. A sequence is a GP if Tₙ ÷ Tₙ₋₁ is a constant independent of n.

Sigma results with lower limit 1

$\sum_{r=1}^{n} a = na, \quad \sum_{r=1}^{n} r = \frac{n(n+1)}{2}, \quad \sum_{r=1}^{n} b^r = \frac{b(b^n - 1)}{b - 1}$

All sums run from r = 1 to n. Σ a is the sum of a constant; Σ r is the sum of an AP with first term 1 and common difference 1; Σ bʳ is the sum of a GP with first term b and common ratio b.

Standard results for Σ r² and Σ r³

$\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4} = \left(\sum_{r=1}^{n} r\right)^2$

Sums run from r = 1 to n. These can be used without proof and will be provided in the question if they are needed.

Rules of summation

$\sum_{r=a}^{b} kT_r = k\sum_{r=a}^{b} T_r, \quad \sum_{r=a}^{b} (T_r \pm U_r) = \sum_{r=a}^{b} T_r \pm \sum_{r=a}^{b} U_r, \quad \sum_{r=a}^{b} T_r = \sum_{r=1}^{b} T_r - \sum_{r=1}^{a-1} T_r$

k is a constant and the first two sums run from r = a to b. The third rule gives the sum of the ath to the bth term as (sum of the first b terms) − (sum of the first a − 1 terms). Each rule can be verified by listing the terms.