Checkout JEE MAINS 2022 Question Paper Analysis : Checkout JEE MAINS 2022 Question Paper Analysis :

If Sn denotes the sum of first n terms of an A.P. whose first term is a and Snx/Sx is independent of x, then Sp =

(a) P3

(b) P2a

(c) Pa2

(d) a3

Solution:

Sum of n terms of AP = Sn = (n/2)[2a + (n-1)d]

Here a is the first term and d is the common difference.

Snx = (nx/2)[2a + (nx-1)d)]

Sx = (x/2)[2a + (x-1)d]

Snx/Sx = (nx/2)[2a + (nx-1)d)] / (x/2)[2a + (x-1)d]

= n[2a + (nx-1)d)]/[2a + (x-1)d]

= (2an + n2xd – nd)/(2a + xd – d)

= [(2a-d)n + n2xd]/[(2a-d) + xd]

For Snx/Sx to be independent of x, 2a – d = 0

=> 2a = d

Sp = (p/2)(2a + (p-1)d)

= (p/2)(2a + (p-1)2a)

= (p/2)(2a + 2pa – 2a)

= (p/2)(2pa)

= p2a

Therefore Sp = p2a

Hence option b is the answer.

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