The coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio 5 : 10 : 14. Then, n is equal to

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Solution

The correct option is D 6
Let the three consecutive terms in $(1 + x)^{n + 5}$ be $t_{r}, t_{r + 1}, t_{r} + 2$ having coefficients
$^{n + 5} C_{r - 1},^{n + 5} C_{r},^{n + 5} C_{r + 1}$
Given,
$^{n + 5} C_{r - 1},^{n + 5} C_{r},^{n + 5} C_{r + 1} = 5 : 10 : 14$
$∴ \frac{^{n + 5} C_{r}}{^{n + 5} C_{r - 1}} = \frac{10}{5} a n d \frac{^{n + 5} C_{r + 1}}{^{n + 5} C_{r}} = \frac{14}{10} \Rightarrow \frac{n + 5 - (r - 1)}{r} = 2 a n d \frac{n - r + 5}{r + 1} = \frac{7}{5}$
$\Rightarrow$ n – r + 6 = 2r and 5n – 5r + 25 = 7r + 7
$\Rightarrow$ n + 6 = 3r and 5n + 18 = 12r
$∴ \frac{n + 6}{3} = \frac{5 n + 18}{12}$
$\Rightarrow$ 4n + 24 = 5n + 18
$\Rightarrow$ n = 6

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