Let {left( {1 + x + {x^2}} right)^{20}}left( {2x + 1} right) = {a_0} + {a₁}{x^1} + {a₂}{x^

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7.Binomial Theorem

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Let ${\left( {1 + x + {x^2}} \right)^{20}}\left( {2x + 1} \right) = {a_0} + {a_1}{x^1} + {a_2}{x^2} + ... + {a_{41}}{x^{41}}$ , then $\frac{{{a_0}}}{1} + \frac{{{a_1}}}{2} + .... + \frac{{{a_{41}}}}{{42}}$ is equal to

A

$\left( {\frac{{{2^{21}} - 1}}{{21}}} \right)$

B

$\left( {\frac{{{3^{21}} - 1}}{{21}}} \right)$

C

$\left( {\frac{{{2^{20}} - 1}}{{20}}} \right)$

D

$\left( {\frac{{{3^{20}} - 1}}{{20}}} \right)$

Solution

$\int\limits_0^1 {{{\left( {1 + x + {x^2}} \right)}^{20}}} (2x + 1)dx$

$=\frac{a_{0}}{1}+\frac{a_{1}}{2}+\ldots .+\frac{a_{41}}{42}$

Standard 11

Mathematics

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