Let S n - n C 0 n C 1- n C 1 n C 2- n C n -1 n C n-If Sn-1-Sn-15-4- then sum of all possible values of nn - N is

S_n= ^nC_0 ^nC_1 + ^nC_1 ^nC_2 + ... + ^nC_n 1 ^nC_n = ^2nC_n+1 S_n+1 = ^2n+2C_n+2 ⇒S_n+1/S_n=^2n+2C_n+2/^2nC_n+1=15/4 ⇒(2n+2)(2n+1)/n(n+2)=15/4 ⇒ n^2 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Binomial Theorem for Any Index

Let S n = n C...

Question

Let $S_{n} =^{n} C_{0}^{n} C_{1} +^{n} C_{1}^{n} C_{2} + . . . +^{n} C_{n - 1}^{n} C_{n}$ .
If $\frac{S_{n + 1}}{S_{n}} = \frac{15}{4}$ , then sum of all possible values of $n (n ϵ N)$ is

Open in App

Solution

$S_{n} =^{n} C_{0}^{n} C_{1} +^{n} C_{1}^{n} C_{2} + . . . +^{n} C_{n - 1}^{n} C_{n} =^{2 n} C_{n + 1}$

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

$\Rightarrow \frac{S_{n + 1}}{S_{n}} = \frac{^{2 n + 2} C_{n + 2}}{^{2 n} C_{n + 1}} = \frac{15}{4}$

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

$\Rightarrow n^{2} - 6 n + 8 = 0$

$\Rightarrow n = 2, 4$

$\Rightarrow$ Sum $= 2 + 4 = 6$

Suggest Corrections

Similar questions

Q. IF

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}

, then n =

Q. 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)

Q. Let

A_{n} =^{n} C_{0}^{n} C_{1} +^{n} C_{1}^{n} C_{2} + . . . +^{n} C_{n - 1}^{n} C_{n}

and

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

,then

n

equals

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 ___.

Join BYJU'S Learning Program

Let Sn= nC0 nC1+ nC1 nC2+...+ nCn−1 nCn. If Sn+1Sn=154, then sum of all possible values of n(n ϵ N) is

Sn= nC0 nC1+ nC1 nC2+...+ nCn−1 nCn= 2nCn+1 Sn+1= 2n+2Cn+2 ⇒Sn+1Sn=2n+2Cn+22nCn+1=154 ⇒(2n+2)(2n+1)n(n+2)=154 ⇒n2−6n+8=0 ⇒n=2, 4 ⇒ Sum =2+4=6

Let $S_{n} =^{n} C_{0}^{n} C_{1} +^{n} C_{1}^{n} C_{2} + . . . +^{n} C_{n - 1}^{n} C_{n}$ .
If $\frac{S_{n + 1}}{S_{n}} = \frac{15}{4}$ , then sum of all possible values of $n (n ϵ N)$ is