The sum of (p+q)^{{th}} and (p-q)^{{th}} terms of an AP is equal to

Correct Answer is : Twice the pthp^{{th}}pth term

Correct Answer is : Twice the pthp^{{th}}pth term

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 24 of 120

Marks: +1, -0

The sum of

(p+q)^{\text{th}}

(p-q)^{\text{th}}

terms of an AP is equal to

Solution:

Concept:

The

n^{\text{th}}

term of an arithmetic progression (AP) is

T_n = a + (n-1)d

, where

a

is the first term and

d

is the common difference.

Explanation:

Let

a

be the first term and

d

the common difference of the AP.

The

(p+q)^{\text{th}}

term is

T_{p+q} = a + (p+q-1)d

The

(p-q)^{\text{th}}

term is

T_{p-q} = a + (p-q-1)d

Add these two terms:

T_{p+q} + T_{p-q} = [a + (p+q-1)d] + [a + (p-q-1)d]

Simplify:

= 2a + (p+q-1 + p-q-1)d = 2a + (2p-2)d

Factor out 2:

= 2[a + (p-1)d]

Notice that

[a + (p-1)d]

is exactly the

p^{\text{th}}

term,

T_p

Hence,

T_{p+q} + T_{p-q} = 2 \times T_p

Answer:

Twice the

p^{\text{th}}

term. (Option C)

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