# If m times the mth term of an AP is equal to n times the nth term of same AP , prove that (m+n)th term of the AP is 0.

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by DeveshGoyal

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by DeveshGoyal

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m{ 2a +(m-1)d}=n{2a+(n-1)d}

m{2a+md-d}=n{2a+(n-1)d}

2am+m2d-md+2an+n2d-nd

2am-2an+m2d-n2d-md+nd=0 [m2 and n2 means "m" square and "n" square]

2a(m-n)+d(m2-n2)-d(m-n)+0

(m-n){2a+(m+n-1)d}=0

2a+(m+n-1)d=0

(m+n)th term=0