Calculus Examples

$\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{16}$

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 $\frac{1}{2}$ 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 $a_{1} = \frac{1}{2}$ and $r = \frac{1}{2}$ .

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

Step 4

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{n - 1}$ by adding the exponents.

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Step 4.1

Multiply $\frac{1}{2}$ by ${(\frac{1}{2})}^{n - 1}$ .

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Step 4.1.1

Raise $\frac{1}{2}$ to the power of $1$ .

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

Step 4.1.2

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

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

Step 4.2

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

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Step 4.2.1

Subtract $1$ from $1$ .

$a_{n} = {(\frac{1}{2})}^{n + 0}$

Step 4.2.2

Add $n$ and $0$ .

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

Step 5

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

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

Step 6

One to any power is one.

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