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Question

If the sum of first m terms of an A.P is same as the sum of its first n terms (mn), show that the sum of its first (m+n) terms is zero

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Solution

Let the AP be denoted as

a1,a2,a3,,an,

with a common difference d.

Sum of first m terms:

Sm=m2(2a1+(m1)d)

Sum of tfirst n terms:

Sn=n2(2a1+(n1)d)

Given that the 2 sums are equal. Hence,

m2(2a1+(m1)d) =n2(2a1+(n1)d)

2a1m+(m2m)d=2a1n+(n2n)d

2a1=d(1mn)

Sum of first (m+n) terms is given by

Sm+n=m+n2[2a1+(m+n1)d]

Substituting
2a1=d(1mn),

Sm+n=m+n2[d(1mn)+(m+n1)d]

=0

Hence, proved that the sum of the first (m+n) terms is 0.

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