Calculus Examples

20

$20$ ,

4

$4$ ,

\frac{4}{5}

$\frac{4}{5}$

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{1}{5}

$\frac{1}{5}$ gives the next term. In other words,

a_{n} = a_{1} r^{n - 1}

$a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence:

r = \frac{1}{5}

$r = \frac{1}{5}$

Step 2

The sum of a series

S_{n}

$S_{n}$ is calculated using the formula

S_{n} = \frac{a (1 - r^{n})}{1 - r}

$S_{n} = \frac{a (1 - r^{n})}{1 - r}$ . For the sum of an infinite geometric series

S_{\infty}

$S_{\infty}$ , as

n

$n$ approaches

\infty

$\infty$ ,

1 - r^{n}

$1 - r^{n}$ approaches

1

$1$ . Thus,

\frac{a (1 - r^{n})}{1 - r}

$\frac{a (1 - r^{n})}{1 - r}$ approaches

\frac{a}{1 - r}

$\frac{a}{1 - r}$ .

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

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

Step 3

The values

a = 20

$a = 20$ and

r = \frac{1}{5}

$r = \frac{1}{5}$ can be put in the equation

S_{\infty}

$S_{\infty}$ .

S_{\infty} = \frac{20}{1 - \frac{1}{5}}

$S_{\infty} = \frac{20}{1 - \frac{1}{5}}$

Step 4

Simplify the equation to find

S_{\infty}

$S_{\infty}$ .

Tap for more steps...

Step 4.1

Simplify the denominator.

Tap for more steps...

Step 4.1.1

Write

1

$1$ as a fraction with a common denominator.

S_{\infty} = \frac{20}{\frac{5}{5} - \frac{1}{5}}

$S_{\infty} = \frac{20}{\frac{5}{5} - \frac{1}{5}}$

Step 4.1.2

Combine the numerators over the common denominator.

S_{\infty} = \frac{20}{\frac{5 - 1}{5}}

$S_{\infty} = \frac{20}{\frac{5 - 1}{5}}$

Step 4.1.3

Subtract

1

$1$ from

5

$5$ .

S_{\infty} = \frac{20}{\frac{4}{5}}

$S_{\infty} = \frac{20}{\frac{4}{5}}$

S_{\infty} = \frac{20}{\frac{4}{5}}

$S_{\infty} = \frac{20}{\frac{4}{5}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

S_{\infty} = 20 (\frac{5}{4})

$S_{\infty} = 20 (\frac{5}{4})$

Step 4.3

Cancel the common factor of

4

$4$ .

Tap for more steps...

Step 4.3.1

Factor

4

$4$ out of

20

$20$ .

S_{\infty} = 4 (5) (\frac{5}{4})

$S_{\infty} = 4 (5) (\frac{5}{4})$

Step 4.3.2

Cancel the common factor.

S_{\infty} = 4 \cdot (5 (\frac{5}{4}))

$S_{\infty} = 4 \cdot (5 (\frac{5}{4}))$

Step 4.3.3

Rewrite the expression.

S_{\infty} = 5 \cdot 5

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

S_{\infty} = 5 \cdot 5

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

Step 4.4

Multiply

5

$5$ by

5

$5$ .

S_{\infty} = 25

$S_{\infty} = 25$

S_{\infty} = 25

$S_{\infty} = 25$

Enter YOUR Problem