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  • If the coefficients of three consecutive terms in the expansion of <img alt="\mathrm {\left ( 1+x \right )^{n}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%20%7B%5Cleft%20%28%20

If the coefficients of three consecutive terms in the expansion of \mathrm {\left ( 1+x \right )^{n}} are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Option: 1

5481


Option: 2

3654


Option: 3

2436


Option: 4

1817


Answers (1)

best_answer

$$ \begin{aligned} & \frac{{ }^n C_r}{{ }^n C_{r-1}}=5 \quad \frac{{ }^n C_{r+1}}{{ }^n C_r}=4 \\ & \frac{\mathrm{n}-\mathrm{r}+1}{\mathrm{r}}=5 \quad \mathrm{n}=5 \mathrm{r}+4\, \, .....(2). \\ & \mathrm{n}=6 \mathrm{r}-1 \ldots(1) \\ & \therefore \mathrm{n}=29, \mathrm{r}=5 \\ & \text { Coeff of } 4^{\text {th }} \text { term }={ }^{29} \mathrm{C}_3 \\ & =3654 \end{aligned}

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