First slide

Binomial theorem for positive integral Index

Question

$The sum of the co-efficients in the binomial expansion of {(\frac{1}{x} + 2 x)}^{n} is equal to 6561 .$ Then constant term in the expansion is

Moderate

A

$^{8} C_{4}$
B

${4.}^{8} C_{4}$
C

$2^{4} .^{8} C_{4}$
D

$2^{4} .^{6} C_{4}$

Solution

The sum of all the co-efficients in an expansion is obtained by putting $x = 1$ in the expression.
$∴ {(\frac{1}{1} + 2.1)}^{n} = 6561$
$\Rightarrow 3^{n} = 3^{8} \Rightarrow n = 8$
$In {(\frac{1}{x} + 2 x)}^{8}, T_{r + 1} =^{8} C_{r} {(\frac{1}{x})}^{8 - r} {(2 x)}^{r} =^{8} C_{r} x^{r - 8 + r} 2^{r}$
$This is constant if 2 r - 8 = 0 \Rightarrow r = 4$
$∴ constant term = 2^{4 8} C_{4}$

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