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

Let T_r-1, T_r, T_r+1 are three consecutive terms of (1 + x)ⁿ⁺⁵

${\mathrm{T}}_{\mathrm{r}-1}{=}^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-2}(\mathrm{x}{)}^{\mathrm{r}-2},{\mathrm{T}}_{\mathrm{r}}{=}^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-1}{\mathrm{x}}^{\mathrm{r}-1},{\mathrm{T}}_{\mathrm{r}+1}{=}^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}}{\mathrm{x}}^{\mathrm{r}}$,

where, ${ }^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-2}{:}^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-1}{:}^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}}=5:10:14$.

So $\frac{{ }^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-2}}{5}=\frac{{ }^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}-1}}{10}=\frac{{ }^{\mathrm{n}+5}{\mathrm{C}}_{\mathrm{r}}}{14}$

Comparing first two results we have n - 3r = - 9        (1)

Comparing last two results we have 5n - 12r = - 30   (2)

From equation (1) and (2), n = 6