Calculus Examples

$\sum_{i = 3}^{7} 2 {(\frac{2}{3})}^{i - 1}$

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.

Tap for more steps...

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

$r = \frac{2 {(\frac{2}{3})}^{i + 1 - 1}}{2 {(\frac{2}{3})}^{i - 1}}$

Simplify.

Tap for more steps...

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$r = \frac{2 {(\frac{2}{3})}^{i + 1 - 1}}{2 {(\frac{2}{3})}^{i - 1}}$

Rewrite the expression.

$r = \frac{{(\frac{2}{3})}^{i + 1 - 1}}{{(\frac{2}{3})}^{i - 1}}$

Cancel the common factor of ${(\frac{2}{3})}^{i + 1 - 1}$ and ${(\frac{2}{3})}^{i - 1}$ .

Tap for more steps...

Factor ${(\frac{2}{3})}^{i - 1}$ out of ${(\frac{2}{3})}^{i + 1 - 1}$ .

$r = \frac{{(\frac{2}{3})}^{i - 1} {(\frac{2}{3})}^{i + 0 - (i - 1)}}{{(\frac{2}{3})}^{i - 1}}$

Cancel the common factors.

Tap for more steps...

Multiply by $1$ .

$r = \frac{{(\frac{2}{3})}^{i - 1} {(\frac{2}{3})}^{i + 0 - (i - 1)}}{{(\frac{2}{3})}^{i - 1} \cdot 1}$

Cancel the common factor.

$r = \frac{{(\frac{2}{3})}^{i - 1} {(\frac{2}{3})}^{i + 0 - (i - 1)}}{{(\frac{2}{3})}^{i - 1} \cdot 1}$

Rewrite the expression.

$r = \frac{{(\frac{2}{3})}^{i + 0 - (i - 1)}}{1}$

Divide ${(\frac{2}{3})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{2}{3})}^{i + 0 - (i - 1)}$

Simplify each term.

Tap for more steps...

Apply the distributive property.

$r = {(\frac{2}{3})}^{i + 0 - i - - 1}$

Multiply $- 1$ by $- 1$ .

$r = {(\frac{2}{3})}^{i + 0 - i + 1}$

Add $i$ and $0$ .

$r = {(\frac{2}{3})}^{i - i + 1}$

Subtract $i$ from $i$ .

$r = {(\frac{2}{3})}^{0 + 1}$

Add $0$ and $1$ .

$r = {(\frac{2}{3})}^{1}$

Simplify.

$r = \frac{2}{3}$

Step 3

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

Tap for more steps...

Substitute $3$ for $i$ into $2 {(\frac{2}{3})}^{i - 1}$ .

$a = 2 {(\frac{2}{3})}^{3 - 1}$

Simplify.

Tap for more steps...

Subtract $1$ from $3$ .

$a = 2 {(\frac{2}{3})}^{2}$

Apply the product rule to $\frac{2}{3}$ .

$a = 2 \frac{2^{2}}{3^{2}}$

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

$a = 2 \frac{4}{3^{2}}$

Raise $3$ to the power of $2$ .

$a = 2 (\frac{4}{9})$

Multiply $2 (\frac{4}{9})$ .

Tap for more steps...

Combine $2$ and $\frac{4}{9}$ .

$a = \frac{2 \cdot 4}{9}$

Multiply $2$ by $4$ .

$a = \frac{8}{9}$

Step 4

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

$\frac{8}{9} \cdot \frac{1 - {(\frac{2}{3})}^{5}}{1 - \frac{2}{3}}$

Step 5

Simplify.

Tap for more steps...

Multiply the numerator and denominator of the complex fraction by $3$ .

Tap for more steps...

Multiply $\frac{1 - {(\frac{2}{3})}^{5}}{1 - \frac{2}{3}}$ by $\frac{3}{3}$ .

$\frac{8}{9} (\frac{3}{3} \cdot \frac{1 - {(\frac{2}{3})}^{5}}{1 - \frac{2}{3}})$

Combine.

$\frac{8}{9} \cdot \frac{3 (1 - {(\frac{2}{3})}^{5})}{3 (1 - \frac{2}{3})}$

Apply the distributive property.

$\frac{8}{9} \cdot \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{5})}{3 \cdot 1 + 3 (- \frac{2}{3})}$

Cancel the common factor of $3$ .

Tap for more steps...

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{8}{9} \cdot \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{5})}{3 \cdot 1 + 3 (\frac{- 2}{3})}$

Cancel the common factor.

$\frac{8}{9} \cdot \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{5})}{3 \cdot 1 + 3 (\frac{- 2}{3})}$

Rewrite the expression.

$\frac{8}{9} \cdot \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{5})}{3 \cdot 1 - 2}$

Simplify the numerator.

Tap for more steps...

Multiply $3$ by $1$ .

$\frac{8}{9} \cdot \frac{3 + 3 (- {(\frac{2}{3})}^{5})}{3 \cdot 1 - 2}$

Apply the product rule to $\frac{2}{3}$ .

$\frac{8}{9} \cdot \frac{3 + 3 (- \frac{2^{5}}{3^{5}})}{3 \cdot 1 - 2}$

Cancel the common factor of $3$ .

Tap for more steps...

Move the leading negative in $- \frac{2^{5}}{3^{5}}$ into the numerator.

$\frac{8}{9} \cdot \frac{3 + 3 \frac{- 2^{5}}{3^{5}}}{3 \cdot 1 - 2}$

Factor $3$ out of $3^{5}$ .

$\frac{8}{9} \cdot \frac{3 + 3 \frac{- 2^{5}}{3 \cdot 3^{4}}}{3 \cdot 1 - 2}$

Cancel the common factor.

$\frac{8}{9} \cdot \frac{3 + 3 \frac{- 2^{5}}{3 \cdot 3^{4}}}{3 \cdot 1 - 2}$

Rewrite the expression.

$\frac{8}{9} \cdot \frac{3 + \frac{- 2^{5}}{3^{4}}}{3 \cdot 1 - 2}$

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

$\frac{8}{9} \cdot \frac{3 + \frac{- 1 \cdot 32}{3^{4}}}{3 \cdot 1 - 2}$

Raise $3$ to the power of $4$ .

$\frac{8}{9} \cdot \frac{3 + \frac{- 1 \cdot 32}{81}}{3 \cdot 1 - 2}$

Multiply $- 1$ by $32$ .

$\frac{8}{9} \cdot \frac{3 + \frac{- 32}{81}}{3 \cdot 1 - 2}$

Move the negative in front of the fraction.

$\frac{8}{9} \cdot \frac{3 - \frac{32}{81}}{3 \cdot 1 - 2}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$\frac{8}{9} \cdot \frac{3 \cdot \frac{81}{81} - \frac{32}{81}}{3 \cdot 1 - 2}$

Combine $3$ and $\frac{81}{81}$ .

$\frac{8}{9} \cdot \frac{\frac{3 \cdot 81}{81} - \frac{32}{81}}{3 \cdot 1 - 2}$

Combine the numerators over the common denominator.

$\frac{8}{9} \cdot \frac{\frac{3 \cdot 81 - 32}{81}}{3 \cdot 1 - 2}$

Simplify the numerator.

Tap for more steps...

Multiply $3$ by $81$ .

$\frac{8}{9} \cdot \frac{\frac{243 - 32}{81}}{3 \cdot 1 - 2}$

Subtract $32$ from $243$ .

$\frac{8}{9} \cdot \frac{\frac{211}{81}}{3 \cdot 1 - 2}$

Simplify the denominator.

Tap for more steps...

Multiply $3$ by $1$ .

$\frac{8}{9} \cdot \frac{\frac{211}{81}}{3 - 2}$

Subtract $2$ from $3$ .

$\frac{8}{9} \cdot \frac{\frac{211}{81}}{1}$

Divide $\frac{211}{81}$ by $1$ .

$\frac{8}{9} \cdot \frac{211}{81}$

Multiply $\frac{8}{9} \cdot \frac{211}{81}$ .

Tap for more steps...

Multiply $\frac{8}{9}$ by $\frac{211}{81}$ .

$\frac{8 \cdot 211}{9 \cdot 81}$

Multiply $8$ by $211$ .

$\frac{1688}{9 \cdot 81}$

Multiply $9$ by $81$ .

$\frac{1688}{729}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1688}{729}$

Decimal Form:

$2.31550068 \dots$

Mixed Number Form:

$2 \frac{230}{729}$