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What is the approximate sum of this series?

[tex]\[
\sum_{k=1}^8 5\left(\frac{4}{3}\right)^{(k-1)}
\][/tex]

A. [tex]\(\quad 184.77\)[/tex]

B. [tex]\(69.279\)[/tex]

C. [tex]\(0.185\)[/tex]

D. [tex]\(134.83\)[/tex]


Sagot :

To solve the series, we need to evaluate the sum:
[tex]\[ \sum_{k=1}^8 5\left(\frac{4}{3}\right)^{k-1} \][/tex]

This is a geometric series with the first term [tex]\( a = 5 \)[/tex] and the common ratio [tex]\( r = \frac{4}{3} \)[/tex].

The sum of the first [tex]\( n \)[/tex] terms of a geometric series is given by the formula:
[tex]\[ S_n = a \frac{1 - r^n}{1 - r} \][/tex]

Plugging in the given values:
[tex]\[ a = 5, \quad r = \frac{4}{3}, \quad n = 8 \][/tex]

Using these values in the formula, we get:
[tex]\[ S_8 = 5 \frac{1 - \left(\frac{4}{3}\right)^8}{1 - \frac{4}{3}} \][/tex]

Now, evaluating the expression within the series sum involves more detailed computation. After performing the exact calculations, the result of the sum is:
[tex]\[ S_8 \approx 134.83 \][/tex]

Therefore, the correct answer is:
D. 134.83
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