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storchak [24]
3 years ago
7

Which equation could be used to calculate the sum of the geometric series?

Mathematics
1 answer:
Greeley [361]3 years ago
3 0

\dfrac{1}{3}+\dfrac{2}{9}+\dfrac{4}{27}+\dfrac{8}{81}+\dfrac{16}{243} = \\ \\\\ = \sum\limits_{k=1}^{5}\dfrac{2^{k-1}}{3^k} = \sum\limits_{k=1}^{5}\dfrac{2^{k}}{2\cdot 3^k} = \dfrac{1}{2}\cdot \sum\limits_{k=1}^{5}\dfrac{2^{k}}{3^k} = \\ \\\\ = \dfrac{1}{2}\cdot \sum\limits_{k=1}^{5}\Big(\dfrac{2}{3}\Big)^k = \dfrac{1}{2}\cdot \left[\Big(\dfrac{2}{3}\Big)^1+\Big(\dfrac{2}{3}\Big)^2+...+\Big(\dfrac{2}{3}\Big)^5\right] =

= \dfrac{1}{2}\cdot \dfrac{\dfrac{2}{3}\cdot\left[\Big(\dfrac{2}{3}\Big)^5-1\right]}{\dfrac{2}{3}-1} =-\Big(\dfrac{2}{3}\Big)^{5}+1 = \dfrac{-2^5+3^5}{3^5} = \boxed{\dfrac{211}{243}}

\text{I used the geometric series formula for sum: }S_n = \dfrac{b_1\cdot (q^n - 1)}{q-1}

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Step-by-step explanation:

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<h3>Calculation of a triangle's perimeter:</h3>

The triangle's perimeter is equivalent to the sum of its edges, where a, b, and c are its sides.

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With the information provided, the formula will become 'a = 3+2s'

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Triangle perimeter =the total of its edges

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As a result, the garden's perimeter = 3+2s+12+s.

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The variable's coefficient in the abbreviated statement is 3.

Learn more about triangle perimeter:

brainly.com/question/23935199?referrer=searchResults

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