Calculus Examples

$\sum_{i = 1}^{8} 15 + 4^{i - 1}$

Step 1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{8} 15 + 4^{i - 1} = \sum_{k = 1}^{8} 15 + \sum_{k = 1}^{8} 4^{i - 1}$

Step 2

Evaluate $\sum_{k = 1}^{8} 15$ .

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Step 2.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2.2

Substitute the values into the formula.

$(15) (8)$

Step 2.3

Multiply $15$ by $8$ .

$120$

Step 3

Evaluate $\sum_{k = 1}^{8} 4^{i - 1}$ .

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Step 3.1

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 3.2

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

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Step 3.2.1

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

$r = \frac{4^{i + 1 - 1}}{4^{i - 1}}$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Cancel the common factor of $4^{i + 1 - 1}$ and $4^{i - 1}$ .

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Step 3.2.2.1.1

Factor $4^{i - 1}$ out of $4^{i + 1 - 1}$ .

$r = \frac{4^{i - 1} \cdot 4^{i + 0 - (i - 1)}}{4^{i - 1}}$

Step 3.2.2.1.2

Cancel the common factors.

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Step 3.2.2.1.2.1

Multiply by $1$ .

$r = \frac{4^{i - 1} \cdot 4^{i + 0 - (i - 1)}}{4^{i - 1} \cdot 1}$

Step 3.2.2.1.2.2

Cancel the common factor.

$r = \frac{4^{i - 1} \cdot 4^{i + 0 - (i - 1)}}{4^{i - 1} \cdot 1}$

Step 3.2.2.1.2.3

Rewrite the expression.

$r = \frac{4^{i + 0 - (i - 1)}}{1}$

Step 3.2.2.1.2.4

Divide $4^{i + 0 - (i - 1)}$ by $1$ .

$r = 4^{i + 0 - (i - 1)}$

Step 3.2.2.2

Add $i$ and $0$ .

$r = 4^{i - (i - 1)}$

Step 3.2.2.3

Simplify each term.

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Step 3.2.2.3.1

Apply the distributive property.

$r = 4^{i - i - - 1}$

Step 3.2.2.3.2

Multiply $- 1$ by $- 1$ .

$r = 4^{i - i + 1}$

Step 3.2.2.4

Subtract $i$ from $i$ .

$r = 4^{0 + 1}$

Step 3.2.2.5

Add $0$ and $1$ .

$r = 4^{1}$

Step 3.2.2.6

Evaluate the exponent.

$r = 4$

Step 3.3

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

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Step 3.3.1

Substitute $1$ for $i$ into $4^{i - 1}$ .

$a = 4^{1 - 1}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Subtract $1$ from $1$ .

$a = 4^{0}$

Step 3.3.2.2

Anything raised to $0$ is $1$ .

$a = 1$

Step 3.4

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

$1 \frac{1 - 4^{8}}{1 - 1 \cdot 4}$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $\frac{1 - 4^{8}}{1 - 1 \cdot 4}$ by $1$ .

$\frac{1 - 4^{8}}{1 - 1 \cdot 4}$

Step 3.5.2

Simplify the numerator.

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Step 3.5.2.1

Raise $4$ to the power of $8$ .

$\frac{1 - 1 \cdot 65536}{1 - 1 \cdot 4}$

Step 3.5.2.2

Multiply $- 1$ by $65536$ .

$\frac{1 - 65536}{1 - 1 \cdot 4}$

Step 3.5.2.3

Subtract $65536$ from $1$ .

$\frac{- 65535}{1 - 1 \cdot 4}$

Step 3.5.3

Simplify the denominator.

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Step 3.5.3.1

Multiply $- 1$ by $4$ .

$\frac{- 65535}{1 - 4}$

Step 3.5.3.2

Subtract $4$ from $1$ .

$\frac{- 65535}{- 3}$

Step 3.5.4

Divide $- 65535$ by $- 3$ .

$21845$

Step 4

Add the results of the summations.

$120 + 21845$

Step 5

Add $120$ and $21845$ .

$21965$