Calculus Examples

\frac{1}{2}

$\frac{1}{2}$ ,

\frac{1}{4}

$\frac{1}{4}$ ,

\frac{1}{8}

$\frac{1}{8}$ ,

\frac{1}{16}

$\frac{1}{16}$

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}

$\frac{1}{2}$ gives the next term. In other words,

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

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

Geometric Sequence:

r = \frac{1}{2}

$r = \frac{1}{2}$

Step 2

This is the form of a geometric sequence.

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

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

Step 3

Substitute in the values of

a_{1} = \frac{1}{2}

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

r = \frac{1}{2}

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

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

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

Step 4

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

{(\frac{1}{2})}^{n - 1}

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

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Step 4.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

{(\frac{1}{2})}^{n - 1}

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

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Step 4.1.1

Raise

\frac{1}{2}

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

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.2

Combine the opposite terms in

1 + n - 1

$1 + n - 1$ .

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Step 4.2.1

Subtract

1

$1$ from

1

$1$ .

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

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

Step 4.2.2

Add

n

$n$ and

0

$0$ .

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

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

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

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

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

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

Step 5

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 6

One to any power is one.

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

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