IAL 2025 May FP1 Q8

Question

[!problem]

(a) Using the standard summation formulae show that

$\sum_{r=1}^{n} 4r^2 = \frac{2n(n+1)(an+b)}{3}$

where $a$ and $b$ are integers to be determined.

(5)

(b) Prove by induction that, for $n \in \mathbb{Z}^+$

$\sum_{r=1}^{n} r(3r+2) = \frac{n^2(n+1)(9n+7)}{4}$

(5)

Using the results from parts (a) and (b) and showing all stages of your working,

(c) determine the value of $n$ for which

$\sum_{r=1}^{n} r(3r+2) = 15\sum_{r=1}^{2n} (2r+1)$

(Solutions relying entirely on calculator technology are not acceptable.)

(3)