If Sn-nr-1-r-1t-016n nCr rCt 4t - then the value of l where l-n-11-Sn is

S_n= ∑^n_r=1(16^n ^nC_r(∑^r 1_t=0 ^rC_t 4^t ) ) =∑^n_r=1 16^n ^nC_r[(1+4)^r 4^r ] =16^n ∑^n_r=1(^nC_r 5^r ^nC_r 4^r ) =16^n[((1+5)^n 1) ((1+4)^n 1 ) ] = 6 ...

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Byju's Answer

Standard XII

Mathematics

A.G.P

If Sn=∑nr=1∑r...

Question

If $S_{n} = n \sum r = 1 r - 1 \sum t = 0 (\frac{1}{6^{n}}^{n} C_{r}^{r} C_{t} 4^{t}),$ then the value of $l$ where $l = \infty \sum n = 1 (1 - S_{n})$ is

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Solution

The correct option is A $5$
$S_{n} = n \sum r = 1 (\frac{1}{6^{n}}^{n} C_{r} (r - 1 \sum t = 0^{r} C_{t} 4^{t}))$
$= n \sum r = 1 \frac{1}{6^{n}}^{n} C_{r} [(1 + 4)^{r} - 4^{r}]$
$= \frac{1}{6^{n}} n \sum r = 1 (^{n} C_{r} 5^{r} -^{n} C_{r} 4^{r})$
$= \frac{1}{6^{n}} [((1 + 5)^{n} - 1) - ((1 + 4)^{n} - 1)]$
$= \frac{6^{n} - 5^{n}}{6^{n}} = 1 - {(\frac{5}{6})}^{n}$

$\infty \sum n = 1 (1 - S_{n}) = \infty \sum n = 1 {(\frac{5}{6})}^{n} = \frac{\frac{5}{6}}{1 - \frac{5}{6}} = 5$

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Similar questions

Q. If

S_{n} = n \sum r = 1 r - 1 \sum t = 0 (\frac{1}{6^{n}}^{n} C_{r}^{r} C_{t} 4^{t}),

then the value of

l,

where

l = \infty \sum n = 1 (1 - S_{n})

Q. If

S_{n} = \sum_{n}^{r = 0} \frac{1}{^{n} C_{r}}

and

t_{n} = \sum_{n}^{r = 0} \frac{r}{^{n} C_{r}}

, then

\frac{t_{n}}{S_{n}}

is equal to

Q. If

S_{n} = n \sum r = 0 \frac{1}{^{n} C_{r}}

and

P_{n} = n \sum r = 0 \frac{r}{^{n} C_{r}}

then

\frac{S_{n}}{P_{n}}

I f s_{n} = \sum_{r = 0}^{n} \frac{1}{n_{C_{r}}} a n d t_{n} = \sum_{r = 0}^{n} \frac{r}{n_{C_{r}}}, t h e n \frac{t_{n}}{s_{n}} =

Q. If

S_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}}

and

t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}}

then

\frac{t_{n}}{S_{n}}

is equal to

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If Sn=n∑r=1r−1∑t=0(16n nCr rCt 4t), then the value of l where l=∞∑n=1(1−Sn) is

The correct option is A 5Sn=n∑r=1(16n nCr(r−1∑t=0 rCt 4t)) =n∑r=1 16n nCr[(1+4)r−4r] =16n n∑r=1(nCr 5r−nCr 4r) =16n[((1+5)n−1)−((1+4)n−1)] =6n−5n6n=1−(56)n ∞∑n=1(1−Sn)=∞∑n=1(56)n=561−56=5

If $S_{n} = n \sum r = 1 r - 1 \sum t = 0 (\frac{1}{6^{n}}^{n} C_{r}^{r} C_{t} 4^{t}),$ then the value of $l$ where $l = \infty \sum n = 1 (1 - S_{n})$ is