if m times the mth term of an ap is equal to N times the nth term and M is not equal to n show that iit (M + n) term is zero
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according to question
m/2 * (2a + (m-1)d) = n/2 * (2a + (n-1)d)
cutting 2
we get
m(2a + (m-1)d) - n(2a+ (n-1)d) = 0
2am + m^2d - md -2an -n^2d +nd =0
2a(m-n) + (m^2 - n^2)d -(m - n)d =0
2a(m-n) + ( (m + n) (m- n) ) d - (m - n )d = 0
taking (m-n) common
2a + ( m + n -1) d = 0------------ 1
S m+n = m+n/2( 2a + (m+n -1)d
we know that 2a + (m+n)d is 0 from eqn. 1
therefore S m+n = 0
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