Question 0513

Sigma Notation
2013 Paper 1 Question 9 Variant

Question

(i)
It is given that f(r)=r4+2r3+r2+12.f(r) \allowbreak {= r^4 + 2 r^3 + r^2 + 12.} Show that
f(r)−f(r−1)=ar3,f(r) - f(r-1) =ar^{3},
for a constant a{a} to be determined.
Hence find a formula for ∑r=1nr3,{\displaystyle \sum_{r=1}^n r^{3},} fully factorizing your answer.
[5]
(ii)
Given further that
∑r=1nr2(r2+1)=130n(n+1)(2n+1)(3n2+3n+4),\sum_{r=1}^n r^2(r^2+1) = {\textstyle \frac{1}{30}} n(n+1)(2n+1)(3n^2+3n+4),
find ∑r=1nf(r).{\displaystyle \sum_{r=1}^n f(r).}
(You should not simplify your answer.)
[3]

Answer

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