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
m+n2
D
mn

Solution

## The correct option is A 0$$Sm=Sn$$$$Sm=\dfrac{m}{2}\left ( 2a+\left ( m-1 \right ) d\right )$$$$Sn=\dfrac{n}{2}\left [ 2a+\left ( n-1 \right )d \right ]$$$$Sm -Sn =\dfrac{m}{2}\left ( 2a+\left ( m-1 \right ) d\right )-\dfrac{n}{2}\left [ 2a+\left ( n-1 \right )d \right ]= 0$$$$Sm -Sn =\left ( m-n \right )\left ( 2a+\left ( m+n-1 \right )d \right )=0$$ Therefore $$\left ( 2a+\left ( m+n-1 \right )d \right )=0$$$$Sm -Sn =2a\left ( m-n \right )+m\left ( m-1 \right )d-n\left ( n-1 \right )d =0$$$$Sm+n=\dfrac{m+n}{2}\left ( 2a+\left ( \left ( m+n \right )-1 \right )d \right )$$$$2a\left [ m-n \right ]+d\left [ m\left ( m-1 \right ) -n\left ( n-1 \right )\right ]=0$$$$2a\left [m-n \right ]+d\left [ m^{2}-m-n^{2}+n \right ]=0$$$$\left ( m-n \right )\left [ 2a+\left ( m+n-1 \right )d \right ]=0$$$$Sm+n=0$$Option is AMathematics

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