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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
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B
1
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C
m+n2
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D
mn
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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 A



Mathematics

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