Precalculus Examples

$3 + 0.75 + 0.1875 + 0.046875 + \dots$

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 $0.25$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 0.25$

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 $a = 3$ and $r = 0.25$ can be put in the equation $S_{\infty}$ .

$S_{\infty} = \frac{3}{1 - 0.25}$

Step 4

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

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Step 4.1

Subtract $0.25$ from $1$ .

$S_{\infty} = \frac{3}{0.75}$

Step 4.2

Divide $3$ by $0.75$ .

$S_{\infty} = 4$