If {p} denotes the fractional part of the number p, then
$\left\{ {{{{3^{200}}} \over 8}} \right\}$, is equal to :
Solution
$\left\{ {{{{3^{200}}} \over 8}} \right\}$
<br><br>= $\left\{ {{{{{\left( {{3^2}} \right)}^{100}}} \over 8}} \right\}$
<br><br>= $\left\{ {{{{{\left( {1 + 8} \right)}^{100}}} \over 8}} \right\}$
<br><br>= $$\left\{ {{{1 + {}^{100}{C_1}.8 + {}^{100}{C_2}{{.8}^2} + .... + {}^{100}{C_{100}}{{.8}^{100}}} \over 8}} \right\}$$
<br><br>= $\left\{ {{{1 + 8K} \over 8}} \right\}$
<br><br>= $\left\{ {{1 \over 8} + K} \right\}$ where K $\in$ Integer
<br><br>$\therefore$ Fractional part = ${{1 \over 8}}$
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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