First slide

Binomial theorem for positive integral Index

Question

$f {p}$ denotes the fractional part of the number $p$ , then $\{\frac{3^{200}}{8}\}$ is equal to

Moderate

A

$\frac{1}{8}$
B

$\frac{3}{8}$
C

$\frac{7}{8}$
D

$\frac{5}{8}$

Solution

$\begin{array}{l} 3^{200} = {(3^{2})}^{100} = (1 + 8)^{100} \\ \Rightarrow 3^{200} = 1 +^{100} C_{1} \times 8 +^{100} C_{2} \times 8^{2} + \dots +^{100} C_{100} \times 8^{100} \\ \Rightarrow \frac{3^{200}}{8} = \frac{1}{8} +^{100} C_{1} +^{100} C_{2} \times 8 + \dots +^{100} C_{100} \times 8^{99} \end{array}$
$\Rightarrow \frac{3^{200}}{8} = \frac{1}{8} +$ a natural number
$\Rightarrow \{\frac{3^{200}}{8}\} = \frac{1}{8}$

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