Relation between Coefficient and Indices of x and y
If the coeffi...
Question
If the coefficients of rth,(r+1)thand(r+2)th terms in the expansion of (1+x)14 are in A.P., then value r can be
A
5
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B
12
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C
10
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D
9
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Solution
The correct option is D9 Coefficients of rth,(r+1)thand(r+2)th terms are 14Cr−1,14Crand14Cr+1
If these coefficients are in A.P., then 2(14Cr)=14Cr−1+14Cr+1 ⇒2(14)!r!(14−r)!=(14)!(r−1)!(15−r)!+(14)!(r+1)!(13−r)! ⇒2(14)!r!(14−r)!=(14)![(r+1)r+(15−r)(14−r)(r+1)!(15−r)! ⇒2(15−r)(r+1)=2r2−28r+210 ⇒r2−14r+45=0 ⇒(r−5)(r−9)=0 ⇒r=5or9