If in an A.P., Sn= n2p and Sm=m2p, then Sp=?

Here is the answer to your question.

In an A.P.,

We have S n = n 2 p

⇒ 2a + ( n 1 )d = 2np ... (1)

 

Also, S m = m 2 p

⇒ 2a + ( m 1) d = 2mp ... (2)

 

(1) (2)

2a + ( n 1 )d – (2a + ( m 1) d) = 2np – 2mp

nd – d – md + d = 2p ( n – m)

(n m) d = 2p ( n m)

d = 2p

Put value of d in equation (1):

2a + ( n – 1 ) 2p = 2np

a + ( n –1 )p = 2np

a = np – ( n – 1 ) p

= np np + p

⇒ a = p

Now s p

= p 2( p)

= p 3

Thus , S p = p 3

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