First slide

Binomial theorem for positive integral Index

Question

Let $R = (\sqrt{2} + 1)^{2 n + 1}, n \in N,$ and $f = R - [R],$ where [ ]
denote the greatest integer function, $R f$ is equal to

Moderate

A

1
B

$2^{2 n + 1}$
C

$2^{2 n} - 1$
D

none of these

Solution

Let $F = (\sqrt{2} - 1)^{2 n + 1} .$ Note that $0 < F < 1$
Also, $R - F = 2 m$ where
$m = 2 [^{2 n + 1} C_{1} (\sqrt{2})^{2 n} +^{2 n + 1} C_{3} (\sqrt{2})^{2 n - 2} +$ $\dots +^{2 n + 1} C_{2 n + 1}]$ is an integer
$\Rightarrow [R] + f - F = 2 m \Rightarrow f - F = 2 m - [R]$ is an integer
But $- 1 < f - F < 1$ . Thus, $f - F = 0$
$∴ R f = R F = 1$

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