Giải bài 2.15 trang 37

Đề bài

Tính tổng sau đây:

$C_{2021}^{0} - 2 C_{2021}^{1} + 2^{2} C_{2021}^{2} - 2^{3} C_{2021}^{3} + . . . - 2^{2021} C_{2021}^{2021}$

Lời giải chi tiết

Ta có:

$\begin{array}{l} (x - 2)^{2021} = C_{2021}^{0} x^{2021} + C_{2021}^{1} x^{2020} (- 2) + C_{2021}^{2} x^{2019} (- 2)^{2} + C_{2021}^{3} x^{2018} (- 2)^{3} + . . . + C_{2021}^{2021} (- 2)^{2021} \\ = C_{2021}^{0} x^{2021} - 2 C_{2021}^{1} x^{2020} + 2^{2} C_{2021}^{2} x^{2019} - 2^{3} C_{2021}^{3} x^{2018} + . . . - 2^{2021} C_{2021}^{2021} \end{array}$

Thay $x = 1$ vào cả hai vế, ta suy ra

$C_{2021}^{0} - 2 C_{2021}^{1} + 2^{2} C_{2021}^{2} - 2^{3} C_{2021}^{3} + . . . - 2^{2021} C_{2021}^{2021} = (- 1)^{2021} = - 1$