In an arithmetic progression

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

Problem

In an arithmetic progression, the

p

th term is

x

and the

q

th term is

y

, where

p

and

q

are positive integers with

p \neq q

. (i) Write the

p

th and

q

th terms of the progression in terms of its first term

a

and common difference

d

. (ii) Using the relations obtained in part (i), express

a

and

d

in terms of

p, q, x,

and

y

. (iii) Hence, derive an expression for the sum of the first

(p+q)

terms of the progression in terms of

p, q, x,

and

y

only. (iv) Simplify your expression as far as possible and state any special condition under which the sum becomes independent of one of the variables

p

q

. (No numerical substitution is required.)

Related theory

Arithmetic Progressions
Sequences and Series
Algebra
Sum of first n terms of an AP

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