Find the Sum of the Series 5 , 5/2 , 5/4 , 5/8

$5$ , $\frac{5}{2}$ , $\frac{5}{4}$ , $\frac{5}{8}$

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

Geometric Sequence: $r = \frac{1}{2}$

Step 2

This is the form of a geometric sequence.

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

Step 3

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

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

Step 4

Apply the product rule to $\frac{1}{2}$ .

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

Step 5

One to any power is one.

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

Step 6

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

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

Step 7

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 8

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

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

Step 9

Simplify the numerator.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

Combine $- 1$ and $\frac{16}{16}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $16$ .

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

Subtract $16$ from $1$ .

$S_{4} = 5 \cdot \frac{\frac{- 15}{16}}{\frac{1}{2} - 1}$

Move the negative in front of the fraction.

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1}$

Step 10

Simplify the denominator.

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Combine $- 1$ and $\frac{2}{2}$ .

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Combine the numerators over the common denominator.

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{1 - 1 \cdot 2}{2}}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{1 - 2}{2}}$

Subtract $2$ from $1$ .

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{\frac{- 1}{2}}$

Move the negative in front of the fraction.

$S_{4} = 5 \cdot \frac{- \frac{15}{16}}{- \frac{1}{2}}$

Step 11

Dividing two negative values results in a positive value.

$S_{4} = 5 \cdot \frac{\frac{15}{16}}{\frac{1}{2}}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = 5 \cdot (\frac{15}{16} \cdot 2)$

Step 13

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$S_{4} = 5 \cdot (\frac{15}{2 (8)} \cdot 2)$

Cancel the common factor.

$S_{4} = 5 \cdot (\frac{15}{2 \cdot 8} \cdot 2)$

Rewrite the expression.

$S_{4} = 5 \cdot \frac{15}{8}$

Step 14

Multiply $5 (\frac{15}{8})$ .

Combine $5$ and $\frac{15}{8}$ .

$S_{4} = \frac{5 \cdot 15}{8}$

Multiply $5$ by $15$ .

$S_{4} = \frac{75}{8}$

Find the Sum of the Series 5 , 5/2 , 5/4 , 5/8

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