Question 0513

Sigma Notation
2013 Paper 1 Question 9 Variant

Question

(i)
It is given that f(r)=2r3+3r2+r+18.f(r) \allowbreak {= 2 r^3 + 3 r^2 + r + 18.} Show that
f(r)−f(r−1)=ar2,f(r) - f(r-1) =ar^{2},
for a constant a{a} to be determined.
Hence find a formula for ∑r=1nr2,{\displaystyle \sum_{r=1}^n r^{2},} fully factorizing your answer.
[5]
(ii)
Given further that
∑r=1nr(2r2+1)=12n(n+1)(n2+n+1),\sum_{r=1}^n r(2r^2 +1) = {\textstyle \frac{1}{2}} n(n+1)(n^2+n+1),
find ∑r=1nf(r).{\displaystyle \sum_{r=1}^n f(r).}
(You should not simplify your answer.)
[3]

Answer

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