Question

If the sum of m terms of an Ap is the same as the sum of its n terms, show that the sum of its ( m+n) terms is zero​

Answer 1

hope this helps you and all the best

Answer 2

Here's the proof -:

∵m/2(2a + (m-1)d) = n/2(2a+(n-1)d)

⇒2am + dm(m-1)  =  2an + dn(n-1)

⇒2a(m-n) = dn² - dn - dm² + dm

             =  d(n-m)(m+n) + d(m-n)

⇒2a(m-n) =  -d(m-n)(m+n) + d(m-n)

→2a(m-n) = d(m-n)[1 - n - m]

∴∴2a = d(1 - n - m)

Now using these relation in this

∵ (m+n)/2(2a + (m+n - 1)d)

We get,

⇒ (m+n)/2(d(1 - n - m) + d(m+n - 1)

⇒(m+n)/2( d - dn -dm + dm + dn - d)

→(m+n)/2( 0 )

∴0.

Hence proved.

Cheers!!