Precalculus Examples

$\sum_{k = 1}^{8} \frac{1}{3} + {(\frac{1}{3})}^{k - 1}$

Step 1

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

$\sum_{k = 1}^{8} \frac{1}{3} + {(\frac{1}{3})}^{k - 1} = \sum_{k = 1}^{8} \frac{1}{3} + \sum_{k = 1}^{8} {(\frac{1}{3})}^{k - 1}$

Step 2

Evaluate $\sum_{k = 1}^{8} \frac{1}{3}$ .

Tap for more steps...

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.

$(\frac{1}{3}) (8)$

Step 2.3

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3}$

Step 3

Evaluate $\sum_{k = 1}^{8} {(\frac{1}{3})}^{k - 1}$ .

Tap for more steps...

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.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

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

Tap for more steps...

Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Cancel the common factors.

Tap for more steps...

Step 3.2.2.1.2.1

Multiply by $1$ .

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

Step 3.2.2.1.2.2

Cancel the common factor.

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

Step 3.2.2.1.2.3

Rewrite the expression.

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

Step 3.2.2.1.2.4

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

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

Step 3.2.2.2

Add $k$ and $0$ .

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

Step 3.2.2.3

Simplify each term.

Tap for more steps...

Step 3.2.2.3.1

Apply the distributive property.

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

Step 3.2.2.3.2

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2.4

Subtract $k$ from $k$ .

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

Step 3.2.2.5

Add $0$ and $1$ .

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

Step 3.2.2.6

Simplify.

$r = \frac{1}{3}$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Simplify.

Tap for more steps...

Step 3.3.2.1

Subtract $1$ from $1$ .

$a = {(\frac{1}{3})}^{0}$

Step 3.3.2.2

Apply the product rule to $\frac{1}{3}$ .

$a = \frac{1^{0}}{3^{0}}$

Step 3.3.2.3

Anything raised to $0$ is $1$ .

$a = \frac{1}{3^{0}}$

Step 3.3.2.4

Anything raised to $0$ is $1$ .

$a = \frac{1}{1}$

Step 3.3.2.5

Divide $1$ by $1$ .

$a = 1$

Step 3.4

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

$1 \frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Multiply $\frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$ by $1$ .

$\frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

Multiply $\frac{1 - {(\frac{1}{3})}^{8}}{1 - \frac{1}{3}}$ by $\frac{3}{3}$ .

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

Step 3.5.2.2

Combine.

$\frac{3 (1 - {(\frac{1}{3})}^{8})}{3 (1 - \frac{1}{3})}$

Step 3.5.3

Apply the distributive property.

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

Step 3.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.4.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 3.5.4.2

Cancel the common factor.

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

Step 3.5.4.3

Rewrite the expression.

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

Step 3.5.5

Simplify the numerator.

Tap for more steps...

Step 3.5.5.1

Multiply $3$ by $1$ .

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

Step 3.5.5.2

Apply the product rule to $\frac{1}{3}$ .

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

Step 3.5.5.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.5.3.1

Move the leading negative in $- \frac{1^{8}}{3^{8}}$ into the numerator.

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

Step 3.5.5.3.2

Factor $3$ out of $3^{8}$ .

$\frac{3 + 3 \frac{- 1^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 1}$

Step 3.5.5.3.3

Cancel the common factor.

$\frac{3 + 3 \frac{- 1^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 1}$

Step 3.5.5.3.4

Rewrite the expression.

$\frac{3 + \frac{- 1^{8}}{3^{7}}}{3 \cdot 1 - 1}$

Step 3.5.5.4

One to any power is one.

$\frac{3 + \frac{- 1 \cdot 1}{3^{7}}}{3 \cdot 1 - 1}$

Step 3.5.5.5

Raise $3$ to the power of $7$ .

$\frac{3 + \frac{- 1 \cdot 1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.6

Multiply $- 1$ by $1$ .

$\frac{3 + \frac{- 1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.7

Move the negative in front of the fraction.

$\frac{3 - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.8

To write $3$ as a fraction with a common denominator, multiply by $\frac{2187}{2187}$ .

$\frac{3 \cdot \frac{2187}{2187} - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.9

Combine $3$ and $\frac{2187}{2187}$ .

$\frac{\frac{3 \cdot 2187}{2187} - \frac{1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.10

Combine the numerators over the common denominator.

$\frac{\frac{3 \cdot 2187 - 1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.11

Simplify the numerator.

Tap for more steps...

Step 3.5.5.11.1

Multiply $3$ by $2187$ .

$\frac{\frac{6561 - 1}{2187}}{3 \cdot 1 - 1}$

Step 3.5.5.11.2

Subtract $1$ from $6561$ .

$\frac{\frac{6560}{2187}}{3 \cdot 1 - 1}$

Step 3.5.6

Simplify the denominator.

Tap for more steps...

Step 3.5.6.1

Multiply $3$ by $1$ .

$\frac{\frac{6560}{2187}}{3 - 1}$

Step 3.5.6.2

Subtract $1$ from $3$ .

$\frac{\frac{6560}{2187}}{2}$

Step 3.5.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{6560}{2187} \cdot \frac{1}{2}$

Step 3.5.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.8.1

Factor $2$ out of $6560$ .

$\frac{2 (3280)}{2187} \cdot \frac{1}{2}$

Step 3.5.8.2

Cancel the common factor.

$\frac{2 \cdot 3280}{2187} \cdot \frac{1}{2}$

Step 3.5.8.3

Rewrite the expression.

$\frac{3280}{2187}$

Step 4

Add the results of the summations.

$\frac{8}{3} + \frac{3280}{2187}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

To write $\frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{729}{729}$ .

$\frac{8}{3} \cdot \frac{729}{729} + \frac{3280}{2187}$

Step 5.2

Write each expression with a common denominator of $2187$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.2.1

Multiply $\frac{8}{3}$ by $\frac{729}{729}$ .

$\frac{8 \cdot 729}{3 \cdot 729} + \frac{3280}{2187}$

Step 5.2.2

Multiply $3$ by $729$ .

$\frac{8 \cdot 729}{2187} + \frac{3280}{2187}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{8 \cdot 729 + 3280}{2187}$

Step 5.4

Simplify the numerator.

Tap for more steps...

Step 5.4.1

Multiply $8$ by $729$ .

$\frac{5832 + 3280}{2187}$

Step 5.4.2

Add $5832$ and $3280$ .

$\frac{9112}{2187}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9112}{2187}$

Decimal Form:

$4.16643804 \dots$

Mixed Number Form:

$4 \frac{364}{2187}$