giúp mình với

$c)~C=1+\frac3{2^3}+\frac4{2^4}+\frac5{2^5}+...+\frac{100}{2^{100}}$

Question

giúp mình với

$c)~C=1+\frac3{2^3}+\frac4{2^4}+\frac5{2^5}+...+\frac{100}{2^{100}}$

Ninh Hoàng · Accepted Answer

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$$C=1+\frac{3}{2^3}+\frac{4}{2^4}+\cdots+\frac{100}{2^{100}}$$

$$\frac{1}{2}C=\frac{1}{2}+\frac{3}{2^4}+\frac{4}{2^5}+\cdots+\frac{100}{2^{101}}$$

$$C-\frac{1}{2}C=\left(1+\frac{3}{2^3}+\frac{4}{2^4}+\cdots+\frac{100}{2^{100}}\right)-\left(\frac{1}{2}+\frac{3}{2^4}+\frac{4}{2^5}+\cdots+\frac{100}{2^{101}}\right)$$

$$\frac{1}{2}C=\left(1-\frac{1}{2}\right)+\frac{3}{2^3}+\left(\frac{4}{2^4}-\frac{3}{2^4}\right)+\cdots+\left(\frac{100}{2^{100}}-\frac{99}{2^{100}}\right)-\frac{100}{2^{101}}$$

$$\frac{1}{2}C=\frac{1}{2}+\frac{3}{2^3}+\frac{1}{2^4}+\cdots+\frac{1}{2^{100}}-\frac{100}{2^{101}}$$

$$\frac{1}{2}C=\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots+\frac{1}{2^{100}}\right)-\frac{100}{2^{101}}$$

Đặt $$D=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots+\frac{1}{2^{100}}$$

$$\frac{1}{2}D=\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots+\frac{1}{2^{101}}$$

$$D-\frac{1}{2}D=\left(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^{100}}\right)-\left(\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\cdots+\frac{1}{2^{101}}\right)$$

$$\frac{1}{2}D=\frac{1}{2}-\frac{1}{2^{101}}$$

$$D=\frac{2^{100}-1}{2^{100}}$$

Thay D vào C, ta có:

$$\frac{1}{2}C=\frac{2^{100}-1}{2^{100}}-\frac{100}{2^{101}}$$

$$C=\frac{2^{101}-2-100}{2^{100}}=\frac{2.\left(2^{100}-51\right)}{2^{100}}=\frac{2^{100}-51}{2^{99}}$$.

Timi · Answer

Để tính tổng $ C = 1 + \frac{3}{2^3} + \frac{4}{2^4} + \frac{5}{2^5} + \ldots + \frac{100}{2^{100}} $, chúng ta sẽ sử dụng phương pháp nhân đôi và trừ.

Bước 1: Nhân cả hai vế của tổng $ C $ với 2:
$$ 2C = 2 \left( 1 + \frac{3}{2^3} + \frac{4}{2^4} + \frac{5}{2^5} + \ldots + \frac{100}{2^{100}} \right) $$
$$ 2C = 2 + \frac{3}{2^2} + \frac{4}{2^3} + \frac{5}{2^4} + \ldots + \frac{100}{2^{99}} $$

Bước 2: Trừ $ C $ từ $ 2C $:
$$ 2C - C = \left( 2 + \frac{3}{2^2} + \frac{4}{2^3} + \frac{5}{2^4} + \ldots + \frac{100}{2^{99}} \right) - \left( 1 + \frac{3}{2^3} + \frac{4}{2^4} + \frac{5}{2^5} + \ldots + \frac{100}{2^{100}} \right) $$
$$ C = 2 - 1 + \left( \frac{3}{2^2} - \frac{3}{2^3} \right) + \left( \frac{4}{2^3} - \frac{4}{2^4} \right) + \ldots + \left( \frac{99}{2^{99}} - \frac{99}{2^{100}} \right) - \frac{100}{2^{100}} $$

Bước 3: Đơn giản hóa các cặp số:
$$ C = 1 + \left( \frac{3}{2^2} - \frac{3}{2^3} \right) + \left( \frac{4}{2^3} - \frac{4}{2^4} \right) + \ldots + \left( \frac{99}{2^{99}} - \frac{99}{2^{100}} \right) - \frac{100}{2^{100}} $$
$$ C = 1 + \frac{3}{2^2} - \frac{3}{2^3} + \frac{4}{2^3} - \frac{4}{2^4} + \ldots + \frac{99}{2^{99}} - \frac{99}{2^{100}} - \frac{100}{2^{100}} $$

Bước 4: Kết hợp các số còn lại:
$$ C = 1 + \frac{3}{2^2} - \frac{100}{2^{100}} $$
$$ C = 1 + \frac{3}{4} - \frac{100}{2^{100}} $$
$$ C = 1 + 0.75 - \frac{100}{2^{100}} $$
$$ C = 1.75 - \frac{100}{2^{100}} $$

Vậy, giá trị của $ C $ là:
$$ C = 1.75 - \frac{100}{2^{100}} $$