Calculus Examples

$\sum_{n = 1}^{4} (125) {(\frac{1}{5})}^{n - 1}$

Step 1

Expand the series for each value of $n$ .

$125 {(\frac{1}{5})}^{1 - 1} + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $1$ from $1$ .

$125 {(\frac{1}{5})}^{0} + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.2

Apply the product rule to $\frac{1}{5}$ .

$125 \frac{1^{0}}{5^{0}} + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.3

Anything raised to $0$ is $1$ .

$125 \frac{1}{5^{0}} + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.4

Anything raised to $0$ is $1$ .

$125 (\frac{1}{1}) + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.5

Cancel the common factor of $1$ .

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Step 2.1.5.1

Cancel the common factor.

$125 (\frac{1}{1}) + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.5.2

Rewrite the expression.

$125 \cdot 1 + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.6

Multiply $125$ by $1$ .

$125 + 125 {(\frac{1}{5})}^{2 - 1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.7

Subtract $1$ from $2$ .

$125 + 125 {(\frac{1}{5})}^{1} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.8

Simplify.

$125 + 125 (\frac{1}{5}) + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.9

Cancel the common factor of $5$ .

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Step 2.1.9.1

Factor $5$ out of $125$ .

$125 + 5 (25) \frac{1}{5} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.9.2

Cancel the common factor.

$125 + 5 \cdot 25 \frac{1}{5} + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.9.3

Rewrite the expression.

$125 + 25 + 125 {(\frac{1}{5})}^{3 - 1} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.10

Subtract $1$ from $3$ .

$125 + 25 + 125 {(\frac{1}{5})}^{2} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.11

Apply the product rule to $\frac{1}{5}$ .

$125 + 25 + 125 \frac{1^{2}}{5^{2}} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.12

One to any power is one.

$125 + 25 + 125 \frac{1}{5^{2}} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.13

Raise $5$ to the power of $2$ .

$125 + 25 + 125 (\frac{1}{25}) + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.14

Cancel the common factor of $25$ .

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Step 2.1.14.1

Factor $25$ out of $125$ .

$125 + 25 + 25 (5) \frac{1}{25} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.14.2

Cancel the common factor.

$125 + 25 + 25 \cdot 5 \frac{1}{25} + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.14.3

Rewrite the expression.

$125 + 25 + 5 + 125 {(\frac{1}{5})}^{4 - 1}$

Step 2.1.15

Subtract $1$ from $4$ .

$125 + 25 + 5 + 125 {(\frac{1}{5})}^{3}$

Step 2.1.16

Apply the product rule to $\frac{1}{5}$ .

$125 + 25 + 5 + 125 \frac{1^{3}}{5^{3}}$

Step 2.1.17

One to any power is one.

$125 + 25 + 5 + 125 \frac{1}{5^{3}}$

Step 2.1.18

Raise $5$ to the power of $3$ .

$125 + 25 + 5 + 125 (\frac{1}{125})$

Step 2.1.19

Cancel the common factor of $125$ .

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Step 2.1.19.1

Cancel the common factor.

$125 + 25 + 5 + 125 (\frac{1}{125})$

Step 2.1.19.2

Rewrite the expression.

$125 + 25 + 5 + 1$

Step 2.2

Add $125$ and $25$ .

$150 + 5 + 1$

Step 2.3

Add $150$ and $5$ .

$155 + 1$

Step 2.4

Add $155$ and $1$ .

$156$