Calculus Examples

$\sum_{n = 1}^{6} 2 {(5)}^{n}$

Step 1

The sum of a finite geometric series can be found using the formula $a (\frac{1 - r^{n}}{1 - r})$ where $a$ is the first term and $r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{n + 1}}{a_{n}}$ and simplifying.

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Step 2.1

Substitute $a_{n}$ and $a_{n + 1}$ into the formula for $r$ .

$r = \frac{2 \cdot 5^{n + 1}}{2 \cdot 5^{n}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{2 \cdot 5^{n + 1}}{2 \cdot 5^{n}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{5^{n + 1}}{5^{n}}$

Step 2.2.2

Cancel the common factor of $5^{n + 1}$ and $5^{n}$ .

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Step 2.2.2.1

Factor $5^{n}$ out of $5^{n + 1}$ .

$r = \frac{5^{n} \cdot 5}{5^{n}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{5^{n} \cdot 5}{5^{n} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{5^{n} \cdot 5}{5^{n} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.4

Divide $5$ by $1$ .

$r = 5$

Step 3

Find the first term in the series by substituting in the lower bound and simplifying.

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Step 3.1

Substitute $1$ for $n$ into $2 {(5)}^{n}$ .

$a = 2 \cdot 5^{1}$

Step 3.2

Simplify.

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Step 3.2.1

Evaluate the exponent.

$a = 2 \cdot 5$

Step 3.2.2

Multiply $2$ by $5$ .

$a = 10$

Step 4

Substitute the values of the ratio, first term, and number of terms into the sum formula.

$10 \frac{1 - 5^{6}}{1 - 1 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $5$ to the power of $6$ .

$10 \frac{1 - 1 \cdot 15625}{1 - 1 \cdot 5}$

Step 5.1.2

Multiply $- 1$ by $15625$ .

$10 \frac{1 - 15625}{1 - 1 \cdot 5}$

Step 5.1.3

Subtract $15625$ from $1$ .

$10 \frac{- 15624}{1 - 1 \cdot 5}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $5$ .

$10 \frac{- 15624}{1 - 5}$

Step 5.2.2

Subtract $5$ from $1$ .

$10 (\frac{- 15624}{- 4})$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Factor $2$ out of $10$ .

$2 (5) \frac{- 15624}{- 4}$

Step 5.3.2

Factor $2$ out of $- 4$ .

$2 \cdot 5 \frac{- 15624}{2 \cdot - 2}$

Step 5.3.3

Cancel the common factor.

$2 \cdot 5 \frac{- 15624}{2 \cdot - 2}$

Step 5.3.4

Rewrite the expression.

$5 (\frac{- 15624}{- 2})$

Step 5.4

Combine $5$ and $\frac{- 15624}{- 2}$ .

$\frac{5 \cdot - 15624}{- 2}$

Step 5.5

Multiply $5$ by $- 15624$ .

$\frac{- 78120}{- 2}$

Step 5.6

Divide $- 78120$ by $- 2$ .

$39060$