2 .If the sum of m terms of an AP is the same as the sum of its n term

1.	2 .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 iszero
Answer» Given sum of m terms of and A.P. is the same as the sum of its n terms We know sum of n terms of AP= n/2(2a+(n-1)d)Therefore sum of m terms =m/2(2a+(m-1)d)----(1) sum of n terms=n/2(2a+(n-1)d)-----(2) According to the problem (1)=(2) m/2(2a+(m-1)d)=n/2(2a+(n-1)d) 2ma+(m2-m)d=2na+(n2-n)d (2m-2n)a=(n2-n)d-(m2-m)d 2(m-n)a=n2d-nd-m2d+md 2a(m-n)=-d(m-n)(m+n-1) a=-d/2(m+n-1)----(3) From (3) we got a,sum of m+n terms = m+n/2(2(-d/2(m+n-1)+(m+n-1)d)which is 0 Hence sum of its(m+n) terms is Zero

2 .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 iszero

Answer»

Given sum of m terms of and A.P. is the same as the sum of its n terms

We know sum of n terms of AP= n/2(2a+(n-1)d)Therefore sum of m terms =m/2(2a+(m-1)d)----(1)

sum of n terms=n/2(2a+(n-1)d)-----(2)

According to the problem (1)=(2)

m/2(2a+(m-1)d)=n/2(2a+(n-1)d)

2ma+(m2-m)d=2na+(n2-n)d

(2m-2n)a=(n2-n)d-(m2-m)d

2(m-n)a=n2d-nd-m2d+md

2a(m-n)=-d(m-n)(m+n-1)

a=-d/2(m+n-1)----(3)

From (3) we got a,sum of m+n terms = m+n/2(2(-d/2(m+n-1)+(m+n-1)d)which is 0

Hence sum of its(m+n) terms is Zero