"Let ${}^n{C_r}$ denote the binomial coefficient of xr in the expansion of (1 + x)n. If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $\alpha$ + $\beta$ is equal to ___________."

Question

Accepted Answer

"$\sum\limits_{k = 0}^{10} {({2^2} + 3k){}^{10}{C_k}}$

$$ = 4\sum\limits_{k = 0}^{10} {{}^{10}{C_k}} + 3\sum\limits_{k = 0}^{10} {k.{}^{10}{C_k}} $$

$= 4({2^{10}}) + 3\sum\limits_{k = 0}^{10} {k.{{10} \over k}.{}^9{C_{k - 1}}}$

= $4({2^{10}}) + 3.10({2^9})$

$= 4({2^{10}}) + {3.5.2^{10}}$

$= {2^{10}}(19)$

According to question,

$19({2^{10}}) = \alpha {.3^{10}} + \beta {.2^{10}}$

$\therefore$ $\alpha = 0,\beta = 19$

$\Rightarrow \alpha + \beta = 19$"

Let ${}^n{C_r}$ denote the binomial coefficient of x^r in the expansion of (1 + x)ⁿ. If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $\alpha$ + $\beta$ is equal to ___________.

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Let ${}^n{C_r}$ denote the binomial coefficient of xr in the expansion of (1 + x)n. If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $\alpha$ + $\beta$ is equal to ___________.

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More Binomial Theorem questions

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Let ${}^n{C_r}$ denote the binomial coefficient of x^r in the expansion of (1 + x)ⁿ. If $$\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R$$, then $\alpha$ + $\beta$ is equal to ___________.