Calculus Examples

$\frac{4}{3}$ , $\frac{8}{3}$ , $\frac{16}{3}$ , $\frac{32}{3}$ , $\frac{64}{3}$

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

Geometric Sequence: $r = 2$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{4}{3}$ and $r = 2$ .

$a_{n} = \frac{4}{3} \cdot 2^{n - 1}$

Step 4

Multiply $\frac{4}{3} \cdot 2^{n - 1}$ .

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Step 4.1

Combine $\frac{4}{3}$ and $2^{n - 1}$ .

$a_{n} = \frac{4 \cdot 2^{n - 1}}{3}$

Step 4.2

Rewrite $4$ as $2^{2}$ .

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

Step 4.3

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

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

Step 4.4

Subtract $1$ from $2$ .

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

Step 5

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 6

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

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

Step 7

Simplify the numerator.

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Step 7.1

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

$S_{5} = \frac{4}{3} \cdot \frac{32 - 1}{2 - 1}$

Step 7.2

Subtract $1$ from $32$ .

$S_{5} = \frac{4}{3} \cdot \frac{31}{2 - 1}$

Step 8

Simplify the expression.

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Step 8.1

Subtract $1$ from $2$ .

$S_{5} = \frac{4}{3} \cdot \frac{31}{1}$

Step 8.2

Divide $31$ by $1$ .

$S_{5} = \frac{4}{3} \cdot 31$

Step 9

Multiply $\frac{4}{3} \cdot 31$ .

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Step 9.1

Combine $\frac{4}{3}$ and $31$ .

$S_{5} = \frac{4 \cdot 31}{3}$

Step 9.2

Multiply $4$ by $31$ .

$S_{5} = \frac{124}{3}$