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7.Binomial Theorem
hard
The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+ x )^{ n +2}$, which are in the ratio $1: 3: 5$, is equal to
A
$25$
B
$63$
C
$41$
D
$92$
(JEE MAIN-2023)
Solution
${ }^{n+2} C_{ r -1}:{ }^{n+2} C_{ r }:{ }^{ n +2} C _{ r +1}=1: 3: 5$
$\frac{{ }^{n+2} C_{ r -1}}{{ }^{ n +2} C _{ r }}=\frac{1}{3}$
$n =4 r -3 \ldots \ldots \text { (i) }$
$\frac{{ }^{n+2} C _{ r }}{{ }^{ n +2} C _{ r +1}}=\frac{3}{5}$
$8 r -1=3 n \text {…… (ii) }$
$\text { From, (i) and (ii) }$
$r =2 \text { and } n =5$
$\text { Required sum }=63$
Standard 11
Mathematics
