Define T_n = _ k=1 ^ n (4k+1) - _ k=1 ^ n (2k-3)

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

Problem

Define

T_n = \sum_{k=1}^{n}(4k+1) - \sum_{k=1}^{n}(2k-3).

Work through the following tasks. (i) Simplify

T_n

so that it is expressed solely in terms of

n

without any summation signs. (ii) Show that

T_n

can be written in the form

an^2 + bn + c

, where

a, b, c

are constants to be determined. (iii) Using your result from part (ii), compute

T_{15}

. Give the final answer as an integer.