Two successive terms in the expansion of $(1 + x)^{24}$ , whose coefficients are in the ratio 4:1 are $(r + 1)^{t h}$ and $(r)^{t h}$ ,r <15. Find the value of r

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Solution

Let the terms be $T_{r + 1}$ and $T_{r}$ . The ratio of their coefficient is $\frac{^{24} C_{r}}{^{24} C_{r - 1}}$ . This is of the form $\frac{^{n} C_{r}}{^{n} C_{r - 1}}$ . We know it is equal to $\frac{n - r + 1}{r}$
$\Rightarrow \frac{n - r + 1}{r} = \frac{4}{1} (n = 24)$
$\Rightarrow \frac{24 - r + 1}{r} = \frac{4}{1}$
$\Rightarrow 24 - r + 1 = 4 r$
$\Rightarrow 5 r = 25$
$\Rightarrow r = 5$

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