# 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.

1
by lusiasingh

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by lusiasingh

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

am+md(m-1)=na+nd(n-1)

a(m-n)=d(n²-n-m²+m)

a(m-n)=d(m-n-(m²-n²))

a(m-n)=d(m-n)(1-m-n)

a=d(1-m-n)

(m+n)th term=a+(m+n-1)d=d(1-m-n)+(m+n-1)d

=d(1-m-n+m+n-1)

=d*0=0