# 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.

Let a and d be the first term and the common difference of the A.P. respectively.

Given,

Sum of first p terms =

Sum of first q terms =

Sp = Sq

p [2a + pd – d] = q [2a + qd – d]

2ap + p (p – 1)d = 2aq + q (q – 1)d

2a (p – q) + d [p(p – 1) – q(q – 1)] = 0

2a (p – q) + d [p2 – q2 – (p – q)] = 0

2a (p – q) + d [(p + q)(p – q) – (p – q)] = 0

(p – q) [2a + d (p + q – 1)] = 0

[ 2a + d (p + q – 1)] = 0

Sp+q = 0

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Interactive quiz on inequalities involving AM and GM45 mins
Check Your progress Part 2| Interactive Quiz: Sequence & Series47 mins
Sequence & Series (Lecture 1)56 mins
Sequence & Series (Lecture 5)42 mins
Sequence & Series (Lecture 7)37 mins
Quick Recap lecture of Sequence & Series56 mins
General Term of Miscellaneous progression46 mins
Improve your understanding of AP & GP27 mins
Geometric Progression58 mins
Sequence & Series (Lecture 4)FREE Class
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses