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  • If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of  are in the ratio 5:1

If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1+x)^{n+5} are in the ratio 5:10:14, then the largest coefficient in this expansion is:
Option: 1 462
Option: 2 330
Option: 3 792
Option: 4 252

Answers (1)

best_answer

Let n + 5 = N

\\\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}}}: \mathrm{N}_{\mathrm{C}_{\mathrm{rt}}}=5: 10: 14 \\ \Rightarrow \frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}}}}{\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}}=\frac{\mathrm{N}+1-\mathrm{r}}{\mathrm{r}}=2 \\ \frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}}{\mathrm{~N}_{\mathrm{C}_{\mathrm{r}}}}=\frac{\mathrm{N}-\mathrm{r}}{\mathrm{r}+1}=\frac{7}{5} \\ \Rightarrow \mathrm{r}=4, \mathrm{~N}=11 \\ \Rightarrow(1+\mathrm{x})^{11} \\ \text { Largest coefficient }={ }^{11} \mathrm{C}_{6}=462

Posted by

himanshu.meshram

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