First slide

Binomial theorem for positive integral Index

Question

The greatest integer contained in $(\sqrt{3} + 1)^{6}$ is

Moderate

A

208
B

416
C

415
D

207

Solution

Let $R = (\sqrt{3} + 1)^{6}$ and $R - [R] = f .$
Put $F = (\sqrt{3} - 1)^{6}$ , then $0 < F < 1$ and
$R + F = (\sqrt{3} + 1)^{6} + (\sqrt{3} - 1)^{6}$
$\begin{aligned} = 2 [3^{3} +^{6} C_{2} (3^{2}) +^{6} C_{4} (3) + 1] \\ = 2 [27 + 135 + 45 + 1] = 416 \end{aligned}$
$\Rightarrow f + F = 416 - [R]$ is an integer
Now, show that $f + F = 1$

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