# Algebra II : Geometric Sequences

## Example Questions

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### Example Question #1 : Understanding Geometric Sequences

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 #1 : Geometric Sequences

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

None of the other answers

None of the other answers

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.

Therefore, the answer is none of the other answers.

### Example Question #1 : 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 #1 : Geometric Sequences

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 #2 : Geometric Sequences

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 #1 : Summations And Sequences

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 #2 : 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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