First slide

Binomial theorem for positive integral Index

Question

$The value of greatest term(Numerically) in the expansion of \sqrt{2} {(1 + \frac{1}{\sqrt{2}})}^{10}$

Moderate

A

$\frac{105}{4}$
B

$\frac{105}{4 \sqrt{2}}$
C

$\frac{105 \sqrt{2}}{8}$
D

$\frac{105 \sqrt{2}}{4}$

Solution

$We have n = 10, x = \frac{1}{\sqrt{2}}$
$Now \frac{(n + 1) | x |}{| x | + 1} = \frac{(10 + 1) \times \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} + 1} = \frac{11 \times \frac{1}{\sqrt{2}}}{\frac{1 + \sqrt{2}}{\sqrt{2}}} = \frac{11}{2.414} = 4.55 = 4 + 0.55 = p + F$
$∴ Numerically greatest term is T_{p + 1} = T_{5}$
$Now T_{5} = T_{4 + 1} = {\sqrt{2}}^{10} C_{4} {(\frac{1}{\sqrt{2}})}^{6} = \sqrt{2} \cdot \frac{{}^{10}C_{4}}{(\sqrt{2})^{6}} = \frac{{}^{10}C_{4}}{(\sqrt{2})^{5}} = \frac{210}{4 \sqrt{2}} = \frac{105 \sqrt{2}}{4}$

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