Calculus Examples

$\frac{4}{3}$ , $\frac{16}{3}$ , $\frac{64}{3}$ , $\frac{256}{3}$ , $\frac{1024}{3}$

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 $4$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 4$

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 $r = 4$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $4$ by $4^{n - 1}$ by adding the exponents.

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Step 4.2.1

Multiply $4$ by $4^{n - 1}$ .

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Step 4.2.1.1

Raise $4$ to the power of $1$ .

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

Step 4.2.1.2

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

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

Step 4.2.2

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

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Step 4.2.2.1

Subtract $1$ from $1$ .

$a_{n} = \frac{4^{n + 0}}{3}$

Step 4.2.2.2

Add $n$ and $0$ .

$a_{n} = \frac{4^{n}}{3}$

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 $S_{5}$ .

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

Step 7

Simplify the numerator.

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Step 7.1

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

$S_{5} = \frac{4}{3} \cdot \frac{1024 - 1}{4 - 1}$

Step 7.2

Subtract $1$ from $1024$ .

$S_{5} = \frac{4}{3} \cdot \frac{1023}{4 - 1}$

Step 8

Simplify terms.

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Step 8.1

Subtract $1$ from $4$ .

$S_{5} = \frac{4}{3} \cdot \frac{1023}{3}$

Step 8.2

Cancel the common factor of $3$ .

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Step 8.2.1

Factor $3$ out of $1023$ .

$S_{5} = \frac{4}{3} \cdot \frac{3 (341)}{3}$

Step 8.2.2

Cancel the common factor.

$S_{5} = \frac{4}{3} \cdot \frac{3 \cdot 341}{3}$

Step 8.2.3

Rewrite the expression.

$S_{5} = 4 \cdot \frac{341}{3}$

Step 8.3

Combine $4$ and $\frac{341}{3}$ .

$S_{5} = \frac{4 \cdot 341}{3}$

Step 8.4

Multiply $4$ by $341$ .

$S_{5} = \frac{1364}{3}$