The sum of the series
Question:

The sum of the series $2 \cdot{ }^{20} \mathrm{C}_{0}+5 \cdot{ }^{20} \mathrm{C}_{1}+8 \cdot{ }^{20} \mathrm{C}_{2}+11 \cdot{ }^{.20} \mathrm{C}_{3}$ $+\ldots+62 \cdot{ }^{20} \mathrm{C}_{20}$ is equal to :

  1. (1) $2^{26}$

  2. (2) $2^{25}$

  3. (3) $2^{23}$

  4. (4) $2^{24}$


Correct Option: , 2

Solution:

$2 \cdot{ }^{20} \mathrm{C}_{0}+5 \cdot{ }^{20} \mathrm{C}_{1}+8 \cdot{ }^{20} \mathrm{C}_{2}+\ldots \ldots+62 \cdot{ }^{20} \mathrm{C}_{20}$

$=\sum_{r=0}^{20}(3 r+2){ }^{20} C_{r}=3 \sum_{r=0}^{20} r \cdot{ }^{20} C_{r}+2 \sum_{r=0}^{20}{ }^{20} C_{r}$

$=60 \sum_{r=1}^{20}{ }^{19} C_{n-1}+2 \sum_{r=0}^{20}{ }^{20} C_{r}$

$=60 \times 2^{19}+2 \times 2^{20}=2^{21}[15+1]=2^{25}$

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