Question

EXAMPLE 35 If the sum of m terms of an A.P. is equal to the sum of either the next n terms or the next p
terms, then prove that

$(m + n)( \frac{1}{m} - \frac{1}{p} = (m + p)( \frac{1}{m} - \frac{1}{n} )$
​

Answer 1

Answer:

Step-by-step explanation:

​  $Let's start arbitrarily with the term a(1).Sum of first m terms= 2m​ ×[a(1)+a(m)]= 2m​ [2a(1)+(m−a)d] .............(1)Sum of next n terms= 2n​ [a(m+1)+a(m+n)]= 2n​ [2a(1)+(2m+n−1)d] .............(2)Sum of next p terms= 2p​ [a(m+n+1)+a(m+n+p)]= 2p​ [2a(1)+(2m+2n+p−1)d] .............(3)All of these equal S. So, now we need to eliminate S and a(1) and d.S( n2​ − m2​ )=(m+n)d$