An arithmetic progression has first term and common difference

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

Problem

An arithmetic progression has first term

a

and common difference

d\,(d\neq 0)

. The sums of its terms are represented by

S_n

. (i) Write the general expression for

S_n

in terms of

a

d

and

n

. (ii) Suppose you are told the numerical values of

S_5

and

S_8

. Explain, without actually carrying out the calculations, why these two pieces of information are sufficient to determine unique values of

a

and

d

. (iii) If

S_5 = S_8

, what conclusion can you draw about the value of

d

? Give reasons for your answer.