NCERT Class XI Mathematics - Sequences and Series - Solutions

Question : 24

Total: 106

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Solution:

Let the first term be a and common difference be d.
According to question Sp=Sq
⇒

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

[2a + (q - 1) d]
⇒ 2ap + (p2−p)d = 2aq + (q2−q)d ⇒ 2ap - 2aq + (p2−p)d - (q2−q)d = 0
⇒ 2a (p – q) + d [(p2–q2) – (p – q)] = 0 ⇒ (p – q) [2a + d {(p + q) – 1}] = 0
⇒ 2a + (p + q – 1) d = 0 (Since p – q ≠ 0) ....(i)
∴ Sp+q =

p+q

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

p+q

(0) = 0 [from (i)]

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