Question:
If Sn denote the sum of n terms of an A.P. with first term a and common difference d such that Sx/Skx is independent of x , then
Answer:
Given AP in which
First term = a
Common difference = d
Number of terms = n
Given Sn denotes the sum of n terms
So
Skx = (kx/2){2a + (kx-1)d}
and Sx = (x/2){2a + (x-1)d}
Now
Sx /Skx = [(x/2){2a + (x-1)d}]/[(kx/2){2a + (kx-1)d}]
= [{2a + (x-1)d}]/[k*{2a + (kx-1)d}]
= [2a + xd-d]/[k*{2a + kxd-d}]
= [(2a - d) + xd]/[k*{(2a - d) + kxd}]
If d= 2a
then
Sx /Skx = [(2a -2a) + x*2a]/[k*{(2a-2a) + k*x*2a}]
= (x*2a)/(k2 *x*2a)
= 1/k2
So Sx /Skx is independent of x if d =2a
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