Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is

The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is \frac{-a(p+q) q}{p-1}.

[Hint: Required sum S\textsubscript{p+q} - S\textsubscript{p}]

 

Answers (1)

The sum of first p terms and the first term is given. We have to find next q terms. So, the total terms become p+q.

Hence, sum of all terms minus the sum of first p terms will give the sum of next q terms. However, sum of first p terms is zero, so the sum of next q terms will be same as sum of all terms.

Sum of n terms is given by S_{n}= \left( \frac{n}{2} \right) \left( 2a+ \left( n-1 \right) d \right)

\\ \text { required sum }=S_{p+q}-S_{p}=\frac{p+q}{2}(2 a+(p+q-1) d) \ldots \ldots(i) \\ S_{p}=\frac{p}{2}(2 a+(p-1) d \\ 0=2 a+(p-1) d \\ d=-\frac{2 a}{p-1}

Replacing the value of d in equation (i)

\\ \text {required sum}=\frac{p+q}{2}\left(2 a+(p+q-1)\left(-\frac{2 a}{p-1}\right)\right. \\ =\frac{p+q}{2}\left(2 a-\frac{2 a p+2 a q-2 a}{p-1}\right) \\ =\frac{p+q}{2}(2 a)\left(1-\frac{p-1}{p-1}-\frac{q}{p-1}\right) \\ \text {On simplifying,} \\ =a(p+q)\left(-\frac{q}{p-1}\right)=-\frac{a(p+q) q}{p-1}

Posted by

infoexpert21

View full answer

Similar Questions

Latest Question