Question

If the sum of the first m terms of an arithmetic sequence is same as the sum of

its first n terms, show that the sum of its first ( m +n ) terms is zero​

Answer 1

See the attachment above

Answer 2

Answer:

Let, the first term and common difference of the respective AP be a and d.

Again, let,

Sum of the first m -terms of the AP =Sm ,

Sum of the first n -terms of the AP =Sn

& Sum of the first (m+n) -terms of the AP =Sm+n.

According to the question,

Sm=Sn

⟹m2{2a+(m−1)d}=n2{2a+(n−1)d}

⟹2am+(m2−m)d=2an+(n2−n)d

⟹2a(m−n)+{(m2−n2)−(m−n)}d=0

⟹(m−n){2a+(m+n−1)d}=0

⟹2a+(m+n−1)d=0...(⋆)

[Assuming, m≠n.]

Now,

Sm+n=m+n2{2a+(m+n−1)d}

=m+n2×0[From (⋆).]

∴Sm+n=0.†

Hope, you'll understand..!!

Step-by-step explanation:

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