Find the Sum of the Series 3/2 , 15/4 , 75/8 , 375/16

$\frac{3}{2}$ , $\frac{15}{4}$ , $\frac{75}{8}$ , $\frac{375}{16}$

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

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

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{3}{2}$ and $r = \frac{5}{2}$ .

$a_{n} = \frac{3}{2} {(\frac{5}{2})}^{n - 1}$

Step 4

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

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

Step 5

Combine.

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

Step 6

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

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

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

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

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

$a_{n} = \frac{3 \cdot 5^{n - 1}}{2^{1 + n - 1}}$

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

Subtract $1$ from $1$ .

$a_{n} = \frac{3 \cdot 5^{n - 1}}{2^{n + 0}}$

Add $n$ and $0$ .

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

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 $S_{4}$ .

$S_{4} = \frac{3}{2} \cdot \frac{{(\frac{5}{2})}^{4} - 1}{\frac{5}{2} - 1}$

Step 9

Simplify the numerator.

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{5^{4}}{2^{4}} - 1}{\frac{5}{2} - 1}$

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625}{2^{4}} - 1}{\frac{5}{2} - 1}$

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625}{16} - 1}{\frac{5}{2} - 1}$

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625}{16} - 1 \cdot \frac{16}{16}}{\frac{5}{2} - 1}$

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625}{16} + \frac{- 1 \cdot 16}{16}}{\frac{5}{2} - 1}$

Combine the numerators over the common denominator.

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625 - 1 \cdot 16}{16}}{\frac{5}{2} - 1}$

Simplify the numerator.

Multiply $- 1$ by $16$ .

$S_{4} = \frac{3}{2} \cdot \frac{\frac{625 - 16}{16}}{\frac{5}{2} - 1}$

Subtract $16$ from $625$ .

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{5}{2} - 1}$

Step 10

Simplify the denominator.

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{5}{2} - 1 \cdot \frac{2}{2}}$

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

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}$

Combine the numerators over the common denominator.

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{5 - 1 \cdot 2}{2}}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{5 - 2}{2}}$

Subtract $2$ from $5$ .

$S_{4} = \frac{3}{2} \cdot \frac{\frac{609}{16}}{\frac{3}{2}}$

Step 11

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = \frac{3}{2} \cdot (\frac{609}{16} \cdot \frac{2}{3})$

Step 12

Cancel the common factor of $3$ .

Factor $3$ out of $609$ .

$S_{4} = \frac{3}{2} \cdot (\frac{3 (203)}{16} \cdot \frac{2}{3})$

Cancel the common factor.

$S_{4} = \frac{3}{2} \cdot (\frac{3 \cdot 203}{16} \cdot \frac{2}{3})$

Rewrite the expression.

$S_{4} = \frac{3}{2} \cdot (\frac{203}{16} \cdot 2)$

Step 13

Cancel the common factor of $2$ .

Factor $2$ out of $16$ .

$S_{4} = \frac{3}{2} \cdot (\frac{203}{2 (8)} \cdot 2)$

Cancel the common factor.

$S_{4} = \frac{3}{2} \cdot (\frac{203}{2 \cdot 8} \cdot 2)$

Rewrite the expression.

$S_{4} = \frac{3}{2} \cdot \frac{203}{8}$

Step 14

Multiply $\frac{3}{2} \cdot \frac{203}{8}$ .

Multiply $\frac{3}{2}$ by $\frac{203}{8}$ .

$S_{4} = \frac{3 \cdot 203}{2 \cdot 8}$

Multiply $3$ by $203$ .

$S_{4} = \frac{609}{2 \cdot 8}$

Multiply $2$ by $8$ .

$S_{4} = \frac{609}{16}$

Find the Sum of the Series 3/2 , 15/4 , 75/8 , 375/16

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