Mathematics

Problem

Consider an arbitrary sequence whose

n

th partial sum is denoted by

S_n\,(n\ge 1)

. a. (i) Prove that for

n\ge 2

, the individual term

a_n

satisfies

a_n = S_n - S_{n-1}

. What is the value of

a_1

in terms of

S_1

? b. (ii) Explain why knowing the value of

S_n

for every positive integer

n

is sufficient to determine each term of the sequence uniquely. c. (iii) Describe how the relation

a_n = S_n - S_{n-1}

can be used to decide whether the sequence forms an arithmetic progression, without carrying out any numerical calculation.