Calculus Examples

135

$135$ ,

45

$45$ ,

15

$15$

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}{3}

$\frac{1}{3}$ 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}{3}

$r = \frac{1}{3}$

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} = 135

$a_{1} = 135$ and

r = \frac{1}{3}

$r = \frac{1}{3}$ .

a_{n} = 135 {(\frac{1}{3})}^{n - 1}

$a_{n} = 135 {(\frac{1}{3})}^{n - 1}$

Step 4

Apply the product rule to

\frac{1}{3}

$\frac{1}{3}$ .

a_{n} = 135 \frac{1^{n - 1}}{3^{n - 1}}

$a_{n} = 135 \frac{1^{n - 1}}{3^{n - 1}}$

Step 5

One to any power is one.

a_{n} = 135 \frac{1}{3^{n - 1}}

$a_{n} = 135 \frac{1}{3^{n - 1}}$

Step 6

Combine

135

$135$ and

\frac{1}{3^{n - 1}}

$\frac{1}{3^{n - 1}}$ .

a_{n} = \frac{135}{3^{n - 1}}

$a_{n} = \frac{135}{3^{n - 1}}$

Step 7

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

a_{4} = \frac{135}{3^{(4) - 1}}

$a_{4} = \frac{135}{3^{(4) - 1}}$

Step 8

Simplify the denominator.

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Step 8.1

Subtract

1

$1$ from

4

$4$ .

a_{4} = \frac{135}{3^{3}}

$a_{4} = \frac{135}{3^{3}}$

Step 8.2

Raise

3

$3$ to the power of

3

$3$ .

a_{4} = \frac{135}{27}

$a_{4} = \frac{135}{27}$

a_{4} = \frac{135}{27}

$a_{4} = \frac{135}{27}$

Step 9

Divide

135

$135$ by

27

$27$ .

a_{4} = 5

$a_{4} = 5$

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