Question

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



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Answer 1

Step-by-step explanation:

In this question, it is given that the sum of m terms of an A.P. is the same as the sum of its n terms and we have to show that the sum of its (m + n) terms is zero. Hence, we have proved that the sum of (m + n) terms of this A.P. is zero.

Answer 2

Answer:

Let a be the first term and d be the common difference of the given A.P. Then, Sm=Sn.

⟹2m{2a+(m−1)d}=2n{2a+(n−1)d}

⟹2a(m−n)+{m(m−1)−n(n−1)}d=0

⟹2a(m−n)+{(m2−n2)−(m−n)}d=0

⟹(m−n){2a+(m+n−1)d}=0

⟹2a+(m+n−1)d=0         [∵m−n=0]          ...(i)

Now,

Sm+n=2m+n{2a+(m+n−1)d}

⟹Sm+n=2m+n×0=0               [Using (i)]

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