If Sn - nC0-nC0- nC1-nC1- nC2-nC2 - - - nCn-nCn- then maximum value of - Sn - 1-Sn - iswhere - denotes greatest integer function

Clearly S_n = #160; ^2nC_n ⇒S_n + 1/S_n = ^2n + 2 C_n + 1/^2n C_n = (2n + 2)(2n +1)/(n +1)(n+1) = 2 (2n + 1)/n+1 = 4 2/n + 1 ∴ [ S_n + ...

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Byju's Answer

Standard XII

Mathematics

Binomial Theorem for Any Index

If Sn = nC0...

Question

If $S_{n} =^{n} C_{0} .^{n} C_{0} +^{n} C_{1} .^{n} C_{1} +^{n} C_{2} .^{n} C_{2} + . . . . . +^{n} C_{n} .^{n} C_{n}$ , then maximum value of $[\frac{S_{n + 1}}{S_{n}}]$ is(where [.] denotes greatest integer function)

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Solution

Clearly $S_{n} =^{2 n} C_{n}$
$\Rightarrow \frac{S_{n + 1}}{S_{n}} = \frac{^{2 n + 2} C_{n + 1}}{^{2 n} C_{n}} = \frac{(2 n + 2) (2 n + 1)}{(n + 1) (n + 1)}$
$= \frac{2 (2 n + 1)}{n + 1} = 4 - \frac{2}{n + 1}$
$∴ [\frac{S_{n + 1}}{S_{n}}] = 3$

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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} a n d \frac{S_{n + 1}}{S_{n}} = \frac{15}{4}

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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}

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\frac{S_{n + 1}}{S_{n}} = \frac{15}{4}

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n (n ϵ N)

Let $S_{1}$ = $^{n} C_{0}$ + $^{n} C_{1}$ + $^{n} C_{2}$ ............. $^{n} C_{n}$ and $S_{2}$ = $^{n} C_{0}$ - $^{n} C_{1}$ + $^{n} C_{2}$ ..............+ $(- 1)^{n}$ $^{n} C_{n}$

Find the value of $\frac{S_{1}}{S_{1} + S_{2}}$

___

Let $S_{1}$ = $^{n} C_{0}$ + $^{n} C_{1}$ + $^{n} C_{2}$ ............. $^{n} C_{n}$ and $S_{2}$ = $^{n} C_{0}$ - $^{n} C_{1}$ + $^{n} C_{2}$ ..............+ $(- 1)^{n}$ $^{n} C_{n}$

Find the value of $\frac{S_{1}}{S_{1} + S_{2}}$ is ___.

Q. If

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}

then which of the following is/are correct ?

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If Sn=nC0.nC0+nC1.nC1+nC2.nC2+.....+nCn.nCn, then maximum value of [Sn+1Sn] is(where [.] denotes greatest integer function)

Clearly Sn=2nCn⇒Sn+1Sn=2n+2Cn+12nCn=(2n+2)(2n+1)(n+1)(n+1)=2(2n+1)n+1=4−2n+1∴[Sn+1Sn]=3

If $S_{n} =^{n} C_{0} .^{n} C_{0} +^{n} C_{1} .^{n} C_{1} +^{n} C_{2} .^{n} C_{2} + . . . . . +^{n} C_{n} .^{n} C_{n}$ , then maximum value of $[\frac{S_{n + 1}}{S_{n}}]$ is(where [.] denotes greatest integer function)

Clearly $S_{n} =^{2 n} C_{n}$
$\Rightarrow \frac{S_{n + 1}}{S_{n}} = \frac{^{2 n + 2} C_{n + 1}}{^{2 n} C_{n}} = \frac{(2 n + 2) (2 n + 1)}{(n + 1) (n + 1)}$
$= \frac{2 (2 n + 1)}{n + 1} = 4 - \frac{2}{n + 1}$
$∴ [\frac{S_{n + 1}}{S_{n}}] = 3$