An A.P whose first term is unity and in which the sum of the first half of any even number of terms to that of the second half of the same number of terms is in constant ratio, then find the common difference.

Dear Sanchit

let the common difference is d

let 2n number of terms

Sn =n/2 [ 2 + (n-1)d]

and

S2n =2n/2 [ 2+ (2n-1)d]

given

 Sn /(S2n -Sn) =  constant

or  (S2n -Sn )/Sn =constant = k

 S2n /Sn  -1 =k

 or  S2n /Sn = constant =k +1

  2n/2 [ 2+ (2n-1)d]  /n/2 [ 2 + (n-1)d]  =k +1

  2[ 2+ (2n-1)d]/[ 2 + (n-1)d] = k +1

  d= 2(k-1)/{n(3-k) +k -1}

 since ratio is independent of n so coefficient of n should be zero

 3-k =0

k=3

 so d = 2

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