Question

If the sum of x terms of an Arithmetic Progression (AP) is the same as the sum of its y terms, show
that the sum of its (x + y) terms is zero,​

Answer 1

Answer:

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

m

=S

n

.

⟹

2

m

{2a+(m−1)d}=

2

n

{2a+(n−1)d}

⟹2a(m−n)+{m(m−1)−n(n−1)}d=0

⟹2a(m−n)+{(m

2

−n

2

)−(m−n)}d=0

⟹(m−n){2a+(m+n−1)d}=0

⟹2a+(m+n−1)d=0 [∵m−n



=0] ...(i)

Now,

S

m+n

=

2

m+n

{2a+(m+n−1)d}

⟹S

m+n

=

2

m+n

×0=0 [Using