Sum of Binomial Coefficients of Odd Numbered Terms
In the binomi...
Question
In the binomial expansion of (1+y)n, where n is a natural number, the coefficients of the 5th,6th and 7th terms are in A.P, find n
A
n=7
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B
n=14
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C
n=8
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D
n=16
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Solution
The correct options are An=14 Dn=7 The coefficient of the fifth term of the binomial expansion is nC4 , sixth term of the binomial expansion is nC5 , seventh term of the binomial expansion is nC6 Given that the 5th,6th and the 7th coefficient are in arithmetic progression, Therefore nC4+nC6 = 2nC5 ⇒n!(n−4)!4!+n!(n−6)!6!=2n!(n−5)!5! ⇒n(n−1)(n−2)(n−3)4!+n(n−1)(n−2)(n−3)(n−4)(n−5)6.5.4!−2 ⇒n(n−1)(n−2)(n−3)(n−4)5.4!=0 Taking out n(n−1)(n−2)(n−3)4! common, we get, (n(n−1)(n−2)(n−3)4!)(1+(n−4)(n−5)30−2(n−4)5)=0 Therefore, (1+(n−4)(n−5)30−2(n−4)5)=0 ⇒30+n2−9n+20+30−12n+48=0 ⇒n2−21n+98=0 (n−7)(n−14)=0 So, we get n=7 or 14