Precalculus Examples

$\frac{5}{2}$ , $2$ , $\frac{8}{5}$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $\frac{4}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{4}{5}$

Step 2

The sum of a series $S_{n}$ is calculated using the formula $S_{n} = \frac{a (1 - r^{n})}{1 - r}$ . For the sum of an infinite geometric series $S_{\infty}$ , as $n$ approaches $\infty$ , $1 - r^{n}$ approaches $1$ . Thus, $\frac{a (1 - r^{n})}{1 - r}$ approaches $\frac{a}{1 - r}$ .

$S_{\infty} = \frac{a}{1 - r}$

Step 3

The values $a = \frac{5}{2}$ and $r = \frac{4}{5}$ can be put in the equation $S_{\infty}$ .

$S_{\infty} = \frac{\frac{5}{2}}{1 - \frac{4}{5}}$

Step 4

Simplify the equation to find $S_{\infty}$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$S_{\infty} = \frac{5}{2} \cdot \frac{1}{1 - \frac{4}{5}}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Write $1$ as a fraction with a common denominator.

$S_{\infty} = \frac{5}{2} \cdot \frac{1}{\frac{5}{5} - \frac{4}{5}}$

Step 4.2.2

Combine the numerators over the common denominator.

$S_{\infty} = \frac{5}{2} \cdot \frac{1}{\frac{5 - 4}{5}}$

Step 4.2.3

Subtract $4$ from $5$ .

$S_{\infty} = \frac{5}{2} \cdot \frac{1}{\frac{1}{5}}$

Step 4.3

Simplify.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$S_{\infty} = \frac{5}{2} \cdot (1 \cdot 5)$

Step 4.3.2

Multiply $5$ by $1$ .

$S_{\infty} = \frac{5}{2} \cdot 5$

Step 4.4

Multiply $\frac{5}{2} \cdot 5$ .

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Step 4.4.1

Combine $\frac{5}{2}$ and $5$ .

$S_{\infty} = \frac{5 \cdot 5}{2}$

Step 4.4.2

Multiply $5$ by $5$ .

$S_{\infty} = \frac{25}{2}$