IF Sn=nC0.nC1+nC1.nC2+....+nCn−1.nCnandSnSn+1=415, then n =

(1+x)n=nC0+nC1n+.....+nCnnn(1+n1)n=nC0+nC1n1+.....+nCn(n1)nSn = coefficient of n in (1+n)n(1+n1)n= coefficient of n in nn(1+n)2n= coefficient of nn+1 in (1+n)2n=2nCn+1Sn+1=2n+2Cn+2Snsn+1=4152n+2Cn+22nCn+1=4152nCn+12n+2Cn+2=415(n+2)!n!(2n+2)!×(2n)!(n+1)!(n−1)!=415(n+2)n(2n+2)(2n+1)=41516n2+24n+8=15n2+30nn2−6n+8=0n−4n−2n+8=0n(n−4)−2(n−4)=0n=2,4

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IF S_n = ^nC_0. ^nC_1 + ^nC_1. ^nC_2 + .... + ^nC_{n - 1}. ^nC

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IF $S_{n} =^{n} C_{0} .^{n} C_{1} +^{n} C_{1} .^{n} C_{2} + .... +^{n} C_{n - 1} .^{n} C_{n} an d \frac{S _{n + 1}}{S _{n}} = \frac{15}{4}$ , then n =

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S_{n} =^{n} C_{0} .^{n} C_{1} +^{n} C_{1} .^{n} C_{2} + .... +^{n} C_{n - 1} .^{n} C_{n} an d \frac{S _{n + 1}}{S _{n}} = \frac{15}{4}

, then n =

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Text solutionVerified

(1 + x)^{n} =^{n} C_{0} +^{n} C_{1}^{n} + ..... +^{n} C_{n} n^{n}

(1 + \frac{1}{n})^{n} =^{n} C_{0} +^{n} C_{1} \frac{1}{n} + ..... +^{n} C_{n} (\frac{1}{n})^{n}

S_{n}

= coefficient of n in

(1 + n)^{n} (1 + \frac{1}{n})^{n}

= coefficient of n in

\frac{( 1 + n ) ^{2 n}}{n ^{n}}

= coefficient of

n^{n + 1}

(1 + n)^{2 n}

=^{2 n} C_{n + 1}

S_{n + 1} =^{2 n + 2} C_{n + 2}

\frac{s _{n + 1}}{S _{n}} = \frac{15}{4}

\frac{^{2 n} C _{n + 1}}{^{2 n + 2} C _{n + 2}} = \frac{15}{4} \frac{^{2 n + 2} C _{n + 2}}{^{2 n} C _{n + 1}} = \frac{15}{4}

\frac{( 2 n + 2 )!}{( n + 2 )! n !} \times \frac{( n + 1 )! ( n - 1 )!}{( 2 n )!} = \frac{15}{4}

\frac{( 2 n + 2 ) ( 2 n + 1 )}{( n + 2 ) n} = \frac{15}{4}

16 n^{2} + 24 n + 8 = 15 n^{2} + 30 n

n^{2} - 6 n + 8 = 0

n - 4 n - 2 n + 8 = 0

n (n - 4) - 2 (n - 4) = 0

n = 2, 4

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Question Text	IF $S_{n} =^{n} C_{0} .^{n} C_{1} +^{n} C_{1} .^{n} C_{2} + .... +^{n} C_{n - 1} .^{n} C_{n} an d \frac{S _{n + 1}}{S _{n}} = \frac{15}{4}$ , then n =
Answer Type	Text solution:1
Upvotes	150

IF Sn​=nC0​.nC1​+nC1​.nC2​+....+nCn−1​.nCn​andSn​Sn+1​​=415​, then n =

Text solutionVerified

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IF $S_{n} =^{n} C_{0} .^{n} C_{1} +^{n} C_{1} .^{n} C_{2} + .... +^{n} C_{n - 1} .^{n} C_{n} an d \frac{S _{n + 1}}{S _{n}} = \frac{15}{4}$ , then n =