Let be a fixed real number

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

Problem

Let

c

be a fixed real number. Define

S = \sum_{i=1}^{p}\,\sum_{k=1}^{q} c,

where

p

and

q

are positive integers. (i) Evaluate the inner sum

\displaystyle \sum_{k=1}^{q} c

. (ii) Using the result of part (i), show that

\displaystyle S = p\,q\,c

. (iii) Find the numerical value of

S

when

p=8

q=15

and

c=-3

. (iv) Generalise your result to the triple sum

\displaystyle \sum_{j=1}^{r}\,\sum_{i=1}^{p}\,\sum_{k=1}^{q} c

and state the simplified expression in terms of

p

q

r

and

c