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7. Using the method of S=2times S-S , find the value of 5+10+20+40+ldots +1280 Method: Phương pháp; Value: Giá trị.

Câu hỏi

7. Using the method of S=2times S-S , find the value of 5+10+20+40+ldots +1280
Method: Phương pháp; Value: Giá trị.
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7. Using the method of S=2times S-S , find the value of 5+10+20+40+ldots +1280 Method: Phương pháp; Value: Giá trị.

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Giải pháp

4.3(157 phiếu bầu)
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Thành Tàithầy · Hướng dẫn 5 năm

Trả lời

**Method:**<br /><br />The given series is a geometric series where each term is double the previous term. The general form of a geometric series is \(a + ar + ar^2 + ar^3 + \ldots + ar^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio.<br /><br />In this series:<br />- The first term \(a = 5\).<br />- The common ratio \(r = 2\).<br /><br />The sum of the first \(n\) terms of a geometric series can be calculated using the formula:<br />\[ S = a \frac{r^n - 1}{r - 1} \]<br /><br />However, the method mentioned in the question seems to be a misunderstanding or misinterpretation of the geometric series sum formula. The correct formula should be applied as shown above.<br /><br />**Value:**<br /><br />To find the sum of the series \(5 + 10 + 20 + 40 + \ldots + 1280\), we need to determine the number of terms (\(n\)) in the series.<br /><br />Since the series is geometric with \(a = 5\) and \(r = 2\), we can find the number of terms by solving for \(n\) in the equation:<br />\[ a \cdot r^{n-1} = 1280 \]<br />\[ 5 \cdot 2^{n-1} = 1280 \]<br />\[ 2^{n-1} = \frac{1280}{5} \]<br />\[ 2^{n-1} = 256 \]<br />\[ n-1 = 8 \]<br />\[ n = 9 \]<br /><br />So, there are 9 terms in the series.<br /><br />Now, using the sum formula for a geometric series:<br />\[ S = 5 \frac{2^9 - 1}{2 - 1} \]<br />\[ S = 5 (512 - 1) \]<br />\[ S = 5 \times 511 \]<br />\[ S = 2555 \]<br /><br />Therefore, the value of the series \(5 + 10 + 20 + 40 + \ldots + 1280\) is \(2555\).