The coefficients of three consecutive terms of (1 + x)n + 5 are in the ratio 5: 10 : 14. Then n =

Engineering

Mathematics

Properties of Binomial Coefficients

Question

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

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Let the three terms are T_r, T_r+1, T_r+2 and three coefficients are ^{n + 5}C_{r – 1} , ^{n + 5}C_r , ^{n + 5}C_{r + 1}

Note: $\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{n - r + 1}{r}$

Here, $\frac{^{n + 5} C_{r}}{^{n + 5} C_{r - 1}} = \frac{10}{5} \Rightarrow \frac{n + 5 - r + 1}{r}$ = 2 $\Rightarrow$ n + 5 – r = 2r $\Rightarrow$ n + 6 = 3r…….(1)

and $\frac{^{n + 5} C_{r + 1}}{^{n + 5} C_{r}} = \frac{14}{10}$

$\Rightarrow \frac{n + 5 - r}{r + 1} = \frac{7}{5}$

$\Rightarrow$ 5n + 25 – 5r = 7r + 7 $\Rightarrow$ 5n + 18 = 12r …….(2)

solving (1) and (2),

5n + 18 = 4 × (n + 6)

$\Rightarrow$ n = 6.