Sums of elementary series — A/L Combined Mathematics

Evaluating polynomial series requires standard summation identities for powers of natural numbers: (1) Sum of the first n natural numbers: ∑_r=1ⁿ r = n(n + 1)/2; (2) Sum of squares: ∑_r=1ⁿ r² = n(n + 1)(2n + 1)/6; (3) Sum of cubes: ∑_r=1ⁿ r³ = [n(n + 1)/2]² = (∑ r)². These fundamental identities allow the summation of any series whose r-th term u_r is a polynomial in r of degree up to 3 by expressing u_r in expanded form and applying the linearity of the summation operator.

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

1. General Term Formulation: Finding the r-th term u_r from the given pattern of terms and expanding it as a polynomial in r.
2. Exact Standard Formulae Substitution: Substituting the standard formulae for ∑r³, ∑r², and ∑r accurately into the expanded sum.
3. Factorization of Result: Factoring the resulting algebraic expression into the simplest product form as requested by the examination question.