Question:
If $ |x|<1, $ then the coefficient of $ {{x}^{6}} $ in the expansion of $ {{(1+x+{{x}^{2}})}^{-3}} $ is
If $ |x|<1, $ then the coefficient of $ {{x}^{6}} $ in the expansion of $ {{(1+x+{{x}^{2}})}^{-3}} $ is
Updated On: Jun 7, 2024
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The Correct Option is A
Solution and Explanation
$ {{(1+x+{{x}^{2}})}^{-3}}={{\left[ \frac{1}{(1+x+{{x}^{2}})} \right]}^{3}} $
$={{\left[ \frac{1-x}{1-{{x}^{3}}} \right]}^{3}} $
$={{(1-x)}^{3}}{{(1-{{x}^{3}})}^{-3}} $
$=(1-{{x}^{3}}-3{{x}^{2}}+3x)(1+3{{x}^{3}}+6{{x}^{6}}+.....) $
$ \therefore $ Coefficient of $ {{x}^{6}} $ in $ {{(1+x+{{x}^{2}})}^{-3}} $
$=6-3=3 $
$={{\left[ \frac{1-x}{1-{{x}^{3}}} \right]}^{3}} $
$={{(1-x)}^{3}}{{(1-{{x}^{3}})}^{-3}} $
$=(1-{{x}^{3}}-3{{x}^{2}}+3x)(1+3{{x}^{3}}+6{{x}^{6}}+.....) $
$ \therefore $ Coefficient of $ {{x}^{6}} $ in $ {{(1+x+{{x}^{2}})}^{-3}} $
$=6-3=3 $
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
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- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.