Finite series — A/L Combined Mathematics

A finite series is the sum of terms of a sequence, denoted using sigma notation as S_n = ∑_r=1ⁿ u_r = u₁ + u₂ + ... + u_n. The fundamental linearity properties of summation are: ∑_r=1ⁿ (a u_r + b v_r) = a ∑_r=1ⁿ u_r + b ∑_r=1ⁿ v_r, and ∑_r=1ⁿ c = n c (where c is constant). For an Arithmetic Progression (AP) with first term a and common difference d, u_r = a + (r - 1)d and S_n = n/2 [2a + (n - 1)d] = n/2 (first + last). For a Geometric Progression (GP) with first term a and common ratio R ≠ 1, u_r = a R^r-1 and S_n = a(1 - Rⁿ) / (1 - R).

Check before you answer (A/L Pure Maths Paper 1)

1. Sigma Linearity Application: Splitting composite summation expressions into scalar multiples of standard known sums without algebraic errors.
2. AP / GP Parameter Recognition: Extracting first term, common difference/ratio from given term conditions and setting up simultaneous equations.
3. Index Shift Mechanics: Correctly adjusting limits when summing from r = k to n: ∑_r=kⁿ u_r = ∑_r=1ⁿ u_r - ∑_r=1^k-1 u_r.