Calculus Examples

$\sum_{i = 0}^{5} 5^{i}$

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 $r = \frac{a_{i + 1}}{a_{i}}$ and simplifying.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 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 $0$ for $i$ into $5^{i}$ .

$a = 5^{0}$

Step 3.2

Anything raised to $0$ is $1$ .

$a = 1$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1 - 5^{6}}{1 - 1 \cdot 5}$ by $1$ .

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

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

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

Step 5.2.2

Multiply $- 1$ by $15625$ .

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

Step 5.2.3

Subtract $15625$ from $1$ .

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

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply $- 1$ by $5$ .

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

Step 5.3.2

Subtract $5$ from $1$ .

$\frac{- 15624}{- 4}$

Step 5.4

Divide $- 15624$ by $- 4$ .

$3906$