Let and let be a positive real number

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Let

S_n=\displaystyle \sum_{k=1}^{n}\bigl(3+2k\bigr)

and let

d

be a positive real number. Investigate the growth of such sums as

n

becomes large. (i) Obtain a closed-form expression for

S_n

in terms of

n

. (ii) Show that the limit

\displaystyle \lim_{n\to\infty}\frac{S_n}{n^{2}}

exists and find its value. (iii) Generalise the result of part (ii) by replacing

3+2k

with the general term

a+(k-1)d

(

d>0

). Evaluate

\lim_{n\to\infty}\frac{1}{n^{2}}\sum_{k=1}^{n}\bigl[a+(k-1)d\bigr].