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Home  >>  CBSE XI  >>  Math  >>  Binomial Theorem
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Using Binomial Theorem, indicate which number is larger $(1.1)^{10000}$ or $1000$.

$\begin{array}{1 1} (1.1)^{10000} \gt 1000. \\(1.1)^{10000} = 1000. \\ (1.1)^{10000} \lt 1000. \\None \;of\; these\end{array} $

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1 Answer

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Toolbox:
  • Express the given number as the sum or difference of two numbers whose powers are easier to evaluate. Then use binomial theorem as follows:
  • $(a+b)^n = \large \sum \limits_{k=0}^{n}\; $$^n \large C$$_k \; a^{n-k}b^k =\; ^n \large C$$_0\;a^nb^0 +\; ^n \large C$$_1\;a^{n-1}b^1+.... ^n\large C$$_n\;a^0b^n$, where $b^0 = 1 = a^{n-n}$
$1.1^{10000} = (1+0.1)^{10000} \rightarrow $ We can now use Binomial Theorem to evaluate this.
$(1+0.1)^{10000} = ^{10000} \large C$$_0\;1^{10000-0}0.1^0 +.^{10000} \large C$$_1\;1^{10000-1}0.1^1 +... + ^{10000} \large C$$_{10000}\;1^{10000-10000}0.1^{10000} $
$\qquad = 1+10000(1.1)+.....$
$\qquad = 11001+....$ (where the rest of the terms are also positive$
$\Rightarrow 1.1^{10000} \gt 1000$
answered Apr 10, 2014 by balaji.thirumalai
edited Apr 10, 2014 by balaji.thirumalai
 

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