If $$Sn=$$ $$nC02+$$ $$nC12+$$ $$nC32+$$…$$+$$ $$nCn2$$, then maximum value of $$Sn+1Sn$$ is $$_________$$.(where$$ [.] denotes the greatest integer $$function)

$$Sn=2nCn$$⇒ $$Sn+1Sn=$$ $$2n+2Cn+12ncn=(2n+2)(2n+1)(n+1)(n+1)=2(2n+1)n+1=4$$−$$2n+1$$∴ $$Sn+1Snmax=3$$

Binomial theorem for positive integral Index

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Question

If $S_{n} = {(^{n} C_{0})}^{2} + {(^{n} C_{1})}^{2} + {(^{n} C_{3})}^{2} + \dots + {(^{n} C_{n})}^{2}$ , then maximum value of $[\frac{S_{n + 1}}{S_{n}}]$ is _________.
(where [.] denotes the greatest integer function)

Moderate

Solution

$S_{n} =^{2 n} C_{n}$
$\begin{matrix} \Rightarrow \frac{S_{n + 1}}{S_{n}} = \frac{^{2 n + 2} C_{n + 1}}{2 n_{c_{n}}} = \frac{(2 n + 2) (2 n + 1)}{(n + 1) (n + 1)} \\ = \frac{2 (2 n + 1)}{n + 1} = 4 - \frac{2}{n + 1} \end{matrix}$
$∴ {[\frac{S_{n + 1}}{S_{n}}]}_{max} = 3$

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