# Calculus Examples

, ,

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 gives the next term. In other words, .

Geometric Sequence:

Step 2

This is the form of a geometric sequence.

Step 3

Substitute in the values of and .

Step 4

Apply the product rule to .

Step 5

One to any power is one.

Step 6

Combine and .

Step 7

This is the formula to find the sum of the first terms of the geometric sequence. To evaluate it, find the values of and .

Step 8

Replace the variables with the known values to find .

Step 9

Step 9.1

Apply the product rule to .

Step 9.2

One to any power is one.

Step 9.3

Raise to the power of .

Step 9.4

To write as a fraction with a common denominator, multiply by .

Step 9.5

Combine and .

Step 9.6

Combine the numerators over the common denominator.

Step 9.7

Simplify the numerator.

Step 9.7.1

Multiply by .

Step 9.7.2

Subtract from .

Step 9.8

Move the negative in front of the fraction.

Step 10

Step 10.1

To write as a fraction with a common denominator, multiply by .

Step 10.2

Combine and .

Step 10.3

Combine the numerators over the common denominator.

Step 10.4

Simplify the numerator.

Step 10.4.1

Multiply by .

Step 10.4.2

Subtract from .

Step 10.5

Move the negative in front of the fraction.

Step 11

Dividing two negative values results in a positive value.

Step 12

Multiply the numerator by the reciprocal of the denominator.

Step 13

Step 13.1

Factor out of .

Step 13.2

Cancel the common factor.

Step 13.3

Rewrite the expression.

Step 14

Step 14.1

Factor out of .

Step 14.2

Cancel the common factor.

Step 14.3

Rewrite the expression.

Step 15

Step 15.1

Factor out of .

Step 15.2

Factor out of .

Step 15.3

Cancel the common factor.

Step 15.4

Rewrite the expression.

Step 16

Combine and .

Step 17

Multiply by .

Step 18

Convert the fraction to a decimal.