CJC/2021/JC1/Promo/P1/Q08

(i) The arithmetic progression is grouped into sets of integers, such that the $n^{\text {th }}$ set contains $n$ integers as shown.
${1},{3,5},{7,9,11},{13,15,17,19}, \ldots \ldots$
(a) Find the total number of terms in the first $n$ sets, and hence show that the last term of the $n^{\text {th }}$ set is $n^2+n-1$.

[2]

(b) Find the first term of the $n^{\text {th }}$ set.

[2]

(c) Show that the sum of the terms in the $n^{\text {th }}$ set is $n^3$.

[1]

(ii) Hence, prove that $1^3+2^3+3^3+\ldots \ldots+n^3=\frac{1}{4} n^2(n+1)^2$.

[2]