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Sanchit Gupta Grade: 12
        

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.

7 years ago

Answers : (1)

Badiuddin askIITians.ismu Expert
147 Points
										

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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Badiuddin

7 years ago
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