Question

If the sum of m terms of an AP is the same as the sum of its n terms. The sum of its (m + n) terms is

• A. 0
• B. 1
• C. $\frac{m+n}{2}$
• D. $\sqrt{mn}$

Answer 1

The correct option is A 0

$Sm=Sn$

$Sm=\frac{m}{2}(2a+(m-1)d)$

$Sn=\frac{n}{2}[2a+(n-1)d]$

$Sm-Sn=\frac{m}{2}(2a+(m-1)d)-\frac{n}{2}[2a+(n-1)d]=0$

$Sm-Sn=(m-n)(2a+(m+n-1)d)=0$

Therefore $(2a+(m+n-1)d)=0$

$Sm-Sn=2a(m-n)+m(m-1)d-n(n-1)d=0$

$Sm+n=\frac{m+n}{2}(2a+((m+n)-1)d)$

$2a[m-n]+d[m(m-1)-n(n-1)]=0$

$2a[m-n]+d[m^2-m-n^2+n]=0$

$(m-n)[2a+(m+n-1)d]=0$

$Sm+n=0$

Option is A