Algebra Examples

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

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 $a_{1} = 1$ and $r = \frac{2}{3}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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 7

Replace the variables with the known values to find $S_{4}$ .

$S_{4} = 1 \cdot \frac{{(\frac{2}{3})}^{4} - 1}{\frac{2}{3} - 1}$

Step 8

Multiply $\frac{{(\frac{2}{3})}^{4} - 1}{\frac{2}{3} - 1}$ by $1$ .

$S_{4} = \frac{{(\frac{2}{3})}^{4} - 1}{\frac{2}{3} - 1}$

Step 9

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

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Step 9.1

Multiply $\frac{{(\frac{2}{3})}^{4} - 1}{\frac{2}{3} - 1}$ by $\frac{3}{3}$ .

$S_{4} = \frac{3}{3} \cdot \frac{{(\frac{2}{3})}^{4} - 1}{\frac{2}{3} - 1}$

Step 9.2

Combine.

$S_{4} = \frac{3 ({(\frac{2}{3})}^{4} - 1)}{3 (\frac{2}{3} - 1)}$

Step 10

Apply the distributive property.

$S_{4} = \frac{3 {(\frac{2}{3})}^{4} + 3 \cdot - 1}{3 (\frac{2}{3}) + 3 \cdot - 1}$

Step 11

Cancel the common factor of $3$ .

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Step 11.1

Cancel the common factor.

$S_{4} = \frac{3 {(\frac{2}{3})}^{4} + 3 \cdot - 1}{3 (\frac{2}{3}) + 3 \cdot - 1}$

Step 11.2

Rewrite the expression.

$S_{4} = \frac{3 {(\frac{2}{3})}^{4} + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12

Simplify the numerator.

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Step 12.1

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

$S_{4} = \frac{3 (\frac{2^{4}}{3^{4}}) + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.2

Cancel the common factor of $3$ .

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Step 12.2.1

Factor $3$ out of $3^{4}$ .

$S_{4} = \frac{3 (\frac{2^{4}}{3 \cdot 3^{3}}) + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.2.2

Cancel the common factor.

$S_{4} = \frac{3 (\frac{2^{4}}{3 \cdot 3^{3}}) + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.2.3

Rewrite the expression.

$S_{4} = \frac{\frac{2^{4}}{3^{3}} + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.3

Raise $2$ to the power of $4$ .

$S_{4} = \frac{\frac{16}{3^{3}} + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.4

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

$S_{4} = \frac{\frac{16}{27} + 3 \cdot - 1}{2 + 3 \cdot - 1}$

Step 12.5

Multiply $3$ by $- 1$ .

$S_{4} = \frac{\frac{16}{27} - 3}{2 + 3 \cdot - 1}$

Step 12.6

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$S_{4} = \frac{\frac{16}{27} - 3 \cdot \frac{27}{27}}{2 + 3 \cdot - 1}$

Step 12.7

Combine $- 3$ and $\frac{27}{27}$ .

$S_{4} = \frac{\frac{16}{27} + \frac{- 3 \cdot 27}{27}}{2 + 3 \cdot - 1}$

Step 12.8

Combine the numerators over the common denominator.

$S_{4} = \frac{\frac{16 - 3 \cdot 27}{27}}{2 + 3 \cdot - 1}$

Step 12.9

Simplify the numerator.

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Step 12.9.1

Multiply $- 3$ by $27$ .

$S_{4} = \frac{\frac{16 - 81}{27}}{2 + 3 \cdot - 1}$

Step 12.9.2

Subtract $81$ from $16$ .

$S_{4} = \frac{\frac{- 65}{27}}{2 + 3 \cdot - 1}$

Step 12.10

Move the negative in front of the fraction.

$S_{4} = \frac{- \frac{65}{27}}{2 + 3 \cdot - 1}$

Step 13

Simplify the denominator.

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Step 13.1

Multiply $3$ by $- 1$ .

$S_{4} = \frac{- \frac{65}{27}}{2 - 3}$

Step 13.2

Subtract $3$ from $2$ .

$S_{4} = \frac{- \frac{65}{27}}{- 1}$

Step 14

Reduce the expression by cancelling the common factors.

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Step 14.1

Dividing two negative values results in a positive value.

$S_{4} = \frac{\frac{65}{27}}{1}$

Step 14.2

Divide $\frac{65}{27}$ by $1$ .

$S_{4} = \frac{65}{27}$