# Algebra II : Geometric Sequences

## Example Questions

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### Example Question #1 : Mathematical Relationships And Basic Graphs

Which of the following is a geometric sequence?

Explanation:

A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. Thus, we look for an implicit definition which involves multiplication of the previous term. The only possibility is:

### Example Question #2 : Mathematical Relationships And Basic Graphs

What is the explicit formula for the above sequence? What is the 20th value?

Explanation:

This is a geometric series. The explicit formula for any geometric series is:

, where  is the common ratio and  is the number of terms.

In this instance  and .

Substitute  into the equation to find the 20th term:

### Example Question #3 : Mathematical Relationships And Basic Graphs

What type of sequence is shown below?

Arithmetic

Subtractive

Multiplicative

Geometric

Explanation:

This series is neither geometric nor arithmetic.

A geometric sequences is multiplied by a common ratio () each term.  An arithmetic series adds the same additional amount () to each term.  This series does neither.

Mutiplicative and subtractive are not types of sequences.

### Example Question #4 : Mathematical Relationships And Basic Graphs

Identify the 10th term in the series:

Explanation:

The explicit formula for a geometric series is

In this problem

Therefore:

### Example Question #5 : Mathematical Relationships And Basic Graphs

Which of the following could be the formula for a geometric sequence?

Explanation:

The explicit formula for a geometric series is .

Therefore, is the only answer that works.

### Example Question #6 : Mathematical Relationships And Basic Graphs

Find the 15th term of the following series:

Explanation:

This series is geometric.  The explicit formula for any geometric series is:

Where represents the term,  is the first term, and is the common ratio.

In this series

Therefore the formula to find the 15th term is:

Explanation:

### Example Question #8 : Mathematical Relationships And Basic Graphs

Find the sum for the first 25 terms in the series

Explanation:

Before we add together the first 25 terms, we need to determine the structure of the series. We know the first term is 60. We can find the common ratio r by dividing the second term by the first:

We can use the formula where A is the first term.

The terms we are adding together are so we can plug in :

Common mistakes would involve order of opperations - make sure you do exponents first, then subtract, then multiply/divide based on what is grouped together.

### Example Question #9 : Mathematical Relationships And Basic Graphs

Give the 33rd term of the Geometric Series

[2 is the first term]

Explanation:

First we need to find the common ratio by dividing the second term by the first:

The  term is

,

so the 33rd term will be

.

### Example Question #10 : Mathematical Relationships And Basic Graphs

Find the 19th term of the sequence

[the first term is 7,000]

Explanation:

First find the common ratio by dividing the second term by the first:

Since the first term is , the nth term can be found using the formula

,

so the 19th term is

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