Question-77881

Question Number 77881 by TawaTawa last updated on 11/Jan/20

Commented by TawaTawa last updated on 11/Jan/20

Please help . 14, 15, 16, 17

Commented by abdomathmax last updated on 12/Jan/20

{(1 + x)}^{n} = \sum_{r = 0}^{n} a_{r} x^{r} = \sum_{r = 0}^{n} C_{n}^{r} x^{r} \Rightarrow a_{r} = C_{n}^{r}

b_{r} = 1 + \frac{a_{r}}{a_{r - 1}} = 1 + \frac{C_{n}^{r}}{C_{n}^{r - 1}} = 1 + \frac{\frac{n!}{r! (n - r)!}}{\frac{n!}{(r - 1)! (n - r + 1)!}}

= 1 + \frac{(r - 1)! (n - r + 1)!}{r! (n - r)!} = 1 + \frac{n - r + 1}{r}

= \frac{r + n - r + 1}{r} = \frac{n + 1}{r} \Rightarrow \prod_{r = 1}^{n} b_{r}

= \prod_{r = 1}^{n} \frac{n + 1}{r} = {(n + 1)}^{n} \prod_{r = 1}^{n} \frac{1}{r}

= {(n + 1)}^{n} (1 \times \frac{1}{2} \times \dots . \times \frac{1}{n}) = \frac{{(n + 1)}^{n}}{n!} = \frac{{(101)}^{100}}{100!} \Rightarrow

n = 100 . a n d t h e c o r r e c t a n s w e r i s b .

Commented by TawaTawa last updated on 12/Jan/20

God bless you sir

Commented by TawaTawa last updated on 12/Jan/20

I appreciate .

Commented by abdomathmax last updated on 16/Jan/20

y o u a r e w e l c o m e