Algebra Examples

$\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \frac{16}{243}$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $\frac{2}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{2}{3}$

Step 2

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 3

Substitute in the values of $a_{1} = \frac{1}{3}$ and $r = \frac{2}{3}$ .

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

Step 4

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

$a_{n} = \frac{1}{3} \cdot \frac{2^{n - 1}}{3^{n - 1}}$

Step 5

Combine.

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3 \cdot 3^{n - 1}}$

Step 6

Multiply $3$ by $3^{n - 1}$ by adding the exponents.

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Step 6.1

Multiply $3$ by $3^{n - 1}$ .

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Step 6.1.1

Raise $3$ to the power of $1$ .

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3^{1} \cdot 3^{n - 1}}$

Step 6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3^{1 + n - 1}}$

Step 6.2

Combine the opposite terms in $1 + n - 1$ .

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Step 6.2.1

Subtract $1$ from $1$ .

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3^{n + 0}}$

Step 6.2.2

Add $n$ and $0$ .

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3^{n}}$

Step 7

Multiply $2^{n - 1}$ by $1$ .

$a_{n} = \frac{2^{n - 1}}{3^{n}}$

Step 8

This is the formula to find the sum of the first $n$ terms of the geometric sequence. To evaluate it, find the values of $r$ and $a_{1}$ .

$S_{n} = \frac{a_{1} (r^{n} - 1)}{r - 1}$

Step 9

Replace the variables with the known values to find $S_{5}$ .

$S_{5} = \frac{1}{3} \cdot \frac{{(\frac{2}{3})}^{5} - 1}{\frac{2}{3} - 1}$

Step 10

Simplify the numerator.

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Step 10.1

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

$S_{5} = \frac{1}{3} \cdot \frac{\frac{2^{5}}{3^{5}} - 1}{\frac{2}{3} - 1}$

Step 10.2

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

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32}{3^{5}} - 1}{\frac{2}{3} - 1}$

Step 10.3

Raise $3$ to the power of $5$ .

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32}{243} - 1}{\frac{2}{3} - 1}$

Step 10.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{243}{243}$ .

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32}{243} - 1 \cdot \frac{243}{243}}{\frac{2}{3} - 1}$

Step 10.5

Combine $- 1$ and $\frac{243}{243}$ .

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32}{243} + \frac{- 1 \cdot 243}{243}}{\frac{2}{3} - 1}$

Step 10.6

Combine the numerators over the common denominator.

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32 - 1 \cdot 243}{243}}{\frac{2}{3} - 1}$

Step 10.7

Simplify the numerator.

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Step 10.7.1

Multiply $- 1$ by $243$ .

$S_{5} = \frac{1}{3} \cdot \frac{\frac{32 - 243}{243}}{\frac{2}{3} - 1}$

Step 10.7.2

Subtract $243$ from $32$ .

$S_{5} = \frac{1}{3} \cdot \frac{\frac{- 211}{243}}{\frac{2}{3} - 1}$

Step 10.8

Move the negative in front of the fraction.

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{2}{3} - 1}$

Step 11

Simplify the denominator.

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Step 11.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{2}{3} - 1 \cdot \frac{3}{3}}$

Step 11.2

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

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}$

Step 11.3

Combine the numerators over the common denominator.

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{2 - 1 \cdot 3}{3}}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Multiply $- 1$ by $3$ .

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{2 - 3}{3}}$

Step 11.4.2

Subtract $3$ from $2$ .

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{\frac{- 1}{3}}$

Step 11.5

Move the negative in front of the fraction.

$S_{5} = \frac{1}{3} \cdot \frac{- \frac{211}{243}}{- \frac{1}{3}}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Dividing two negative values results in a positive value.

$S_{5} = \frac{1}{3} \cdot \frac{\frac{211}{243}}{\frac{1}{3}}$

Step 12.2

Multiply the numerator by the reciprocal of the denominator.

$S_{5} = \frac{1}{3} \cdot (\frac{211}{243} \cdot 3)$

Step 12.3

Cancel the common factor of $3$ .

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Step 12.3.1

Factor $3$ out of $\frac{211}{243} \cdot 3$ .

$S_{5} = \frac{1}{3} \cdot (3 \cdot \frac{211}{243})$

Step 12.3.2

Cancel the common factor.

$S_{5} = \frac{1}{3} \cdot (3 \cdot \frac{211}{243})$

Step 12.3.3

Rewrite the expression.

$S_{5} = \frac{211}{243}$