Find the Sum of the Series 2+4/5+8/25+16/125

$2 + \frac{4}{5} + \frac{8}{25} + \frac{16}{125}$

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 $\frac{2}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{2}{5}$

This is the form of a geometric sequence.

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

Substitute in the values of $a_{1} = 2$ and $r = \frac{2}{5}$ .

$a_{n} = 2 {(\frac{2}{5})}^{n - 1}$

Apply the product rule to $\frac{2}{5}$ .

$a_{n} = 2 \frac{2^{n - 1}}{5^{n - 1}}$

Multiply $2 \frac{2^{n - 1}}{5^{n - 1}}$ .

Combine $2$ and $\frac{2^{n - 1}}{5^{n - 1}}$ .

$a_{n} = \frac{2 \cdot 2^{n - 1}}{5^{n - 1}}$

Multiply $2$ by $2^{n - 1}$ by adding the exponents.

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

Raise $2$ to the power of $1$ .

$a_{n} = \frac{2^{1} \cdot 2^{n - 1}}{5^{n - 1}}$

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

$a_{n} = \frac{2^{1 + n - 1}}{5^{n - 1}}$

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

Subtract $1$ from $1$ .

$a_{n} = \frac{2^{n + 0}}{5^{n - 1}}$

Add $n$ and $0$ .

$a_{n} = \frac{2^{n}}{5^{n - 1}}$

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 $a_{1}$ .

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

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

$S_{4} = 2 \cdot \frac{{(\frac{2}{5})}^{4} - 1}{\frac{2}{5} - 1}$

Simplify the numerator.

Apply the product rule to $\frac{2}{5}$ .

$S_{4} = 2 \cdot \frac{\frac{2^{4}}{5^{4}} - 1}{\frac{2}{5} - 1}$

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

$S_{4} = 2 \cdot \frac{\frac{16}{5^{4}} - 1}{\frac{2}{5} - 1}$

Raise $5$ to the power of $4$ .

$S_{4} = 2 \cdot \frac{\frac{16}{625} - 1}{\frac{2}{5} - 1}$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{625}{625}$ .

$S_{4} = 2 \cdot \frac{\frac{16}{625} - 1 \cdot \frac{625}{625}}{\frac{2}{5} - 1}$

Combine $- 1$ and $\frac{625}{625}$ .

$S_{4} = 2 \cdot \frac{\frac{16}{625} + \frac{- 1 \cdot 625}{625}}{\frac{2}{5} - 1}$

Combine the numerators over the common denominator.

$S_{4} = 2 \cdot \frac{\frac{16 - 1 \cdot 625}{625}}{\frac{2}{5} - 1}$

Simplify the numerator.

Multiply $- 1$ by $625$ .

$S_{4} = 2 \cdot \frac{\frac{16 - 625}{625}}{\frac{2}{5} - 1}$

Subtract $625$ from $16$ .

$S_{4} = 2 \cdot \frac{\frac{- 609}{625}}{\frac{2}{5} - 1}$

Move the negative in front of the fraction.

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{2}{5} - 1}$

Simplify the denominator.

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{2}{5} - 1 \cdot \frac{5}{5}}$

Combine $- 1$ and $\frac{5}{5}$ .

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{2}{5} + \frac{- 1 \cdot 5}{5}}$

Combine the numerators over the common denominator.

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{2 - 1 \cdot 5}{5}}$

Simplify the numerator.

Multiply $- 1$ by $5$ .

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{2 - 5}{5}}$

Subtract $5$ from $2$ .

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{\frac{- 3}{5}}$

Move the negative in front of the fraction.

$S_{4} = 2 \cdot \frac{- \frac{609}{625}}{- \frac{3}{5}}$

Dividing two negative values results in a positive value.

$S_{4} = 2 \cdot \frac{\frac{609}{625}}{\frac{3}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = 2 \cdot (\frac{609}{625} \cdot \frac{5}{3})$

Cancel the common factor of $3$ .

Factor $3$ out of $609$ .

$S_{4} = 2 \cdot (\frac{3 (203)}{625} \cdot \frac{5}{3})$

Cancel the common factor.

$S_{4} = 2 \cdot (\frac{3 \cdot 203}{625} \cdot \frac{5}{3})$

Rewrite the expression.

$S_{4} = 2 \cdot (\frac{203}{625} \cdot 5)$

Cancel the common factor of $5$ .

Factor $5$ out of $625$ .

$S_{4} = 2 \cdot (\frac{203}{5 (125)} \cdot 5)$

Cancel the common factor.

$S_{4} = 2 \cdot (\frac{203}{5 \cdot 125} \cdot 5)$

Rewrite the expression.

$S_{4} = 2 \cdot \frac{203}{125}$

Multiply $2 (\frac{203}{125})$ .

Combine $2$ and $\frac{203}{125}$ .

$S_{4} = \frac{2 \cdot 203}{125}$

Multiply $2$ by $203$ .

$S_{4} = \frac{406}{125}$

Find the Sum of the Series 2+4/5+8/25+16/125

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