One of the most important and exam oriented question from arithmetic progression chapter as it was already asked in previous year paper of 2017 in which we have been asked to find the sum of the first mn terms, if mth term of an AP is 1/n and nth term is 1/m.
Book – RS Aggarwal, Class 10, chapter 5C, question no 7.
Let a be first term and d be the common difference of an AP.
mth term=1/n
so,
am=a+(m−1)d=1/n ………(1)
nth term=1/m
an=a+(n−1)d=1/m ……..(2)
Subtract (2) from (1)
(m−1)d−(n−1)d=1/n−1/m
d(n−n)=(m−n)/mn
d=1/mn ………(3)
From (3) and (1), we get
a+(m−1)×1/mn=1/n
⇒a=1/n−(m−1)/mn
a=1/mn
Sum of first k terms =(k/2)[2a+(k−1)d]
Smn=mn/2[2(1/mn)+(mn−1)(1/mn)]
=mn/2[1/mn+1]
=(1+mn)/2.