Consider an arithmetic progression (A.P.) whose first term is and common difference is

MATHEMATICS · Multi-part · CBSE · Mathematics · Arithmetic Progressions · Arithmetic Progressions

Problem

Consider an arithmetic progression (A.P.) whose first term is

a

and common difference is

d

. Let

S_n

denote the sum of its first

n

terms. (i) Prove that

S_n=\dfrac{n}{2}\bigl[2a+(n-1)d\bigr]

. Clearly state every algebraic step used. (ii) Show that the

n

th term

a_n

of the same A.P. can be written as

a_n=S_n-S_{n-1}

. Explain why this relation holds. (iii) Using the result of part (ii), obtain

a_n

in terms of

a

d

and

n

and verify that it matches the well-known expression

a_n=a+(n-1)d

. Discuss briefly how parts (i) and (ii) together justify the consistency of the two formulas for an A.P.