# High School Math : Sequences and Series

## Example Questions

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### Example Question #1 : Sequences And Series

Evaluate:

None of the other answers are correct.

Explanation:

This sum can be determined using the formula for the sum of an infinite geometric series, with initial term  and common ratio :

### Example Question #1 : Sequences And Series

The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?

110

55

210

105

220

220

Explanation:

An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.

Let  denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:

, where d is the common difference between two consecutive terms.

We are given the 4th and 8th terms in the sequence, so we can write the following equations:

.

We now have a system of two equations with two unknowns:

Let us solve this system by subtracting the equation  from the equation . The result of this subtraction is

.

This means that d = 2.5.

Using the equation , we can find the first term of the sequence.

Ultimately, we are asked to find the hundredth term of the sequence.

### Example Question #1 : Sequences And Series

Find the sum, if possible:

Explanation:

The formula for the summation of an infinite geometric series is

,

where  is the first term in the series and  is the rate of change between succesive terms.  The key here is finding the rate, or pattern, between the terms.  Because this is a geometric sequence, the rate is the constant by which each new term is multiplied.

Plugging in our values, we get:

### Example Question #1 : Sequences And Series

Find the sum, if possible:

Explanation:

The formula for the summation of an infinite geometric series is

,

where  is the first term in the series and  is the rate of change between succesive terms in a series

Because the terms switch sign, we know that the rate must be negative.

Plugging in our values, we get:

### Example Question #1 : Understanding Arithmetic And Geometric Series

Find the sum, if possible:

No solution

No solution

Explanation:

The formula for the summation of an infinite geometric series is

,

where  is the first term in the series and  is the rate of change between succesive terms in a series.

In order for an infinite geometric series to have a sum,  needs to be greater than  and less than , i.e. .

Since , there is no solution.

### Example Question #1 : Sequences And Series

Determine the summation notation for the following series:

Explanation:

The series is a geometric series. The summation notation of a geometric series is

,

where  is the number of terms in the series,  is the first term of the series, and  is the common ratio between terms.

In this series,  is  is , and  is . Therefore, the summation notation of this geometric series is:

This simplifies to:

### Example Question #5 : Sequences And Series

Determine the summation notation for the following series:

Explanation:

The series is a geometric series. The summation notation of a geometric series is

,

where  is the number of terms in the series,  is the first term of the series, and  is the common ratio between terms.

In this series,  is  is , and  is . Therefore, the summation notation of this geometric series is:

This simplifies to:

### Example Question #2 : Sequences And Series

Indicate the sum of the following series:

Explanation:

The formula for the sum of an arithmetic series is

,

where  is the first value in the series,  is the number of terms in the series, and  is the difference between sequential terms in the series.

In this problem we have:

Plugging in our values, we get:

### Example Question #1 : Using Sigma Notation

Indicate the sum of the following series:

Explanation:

The formula for the sum of an arithmetic series is

,

where  is the first value in the series,  is the number of terms in the series, and  is the difference between sequential terms in the series.

Here we have:

Plugging in our values, we get:

### Example Question #2051 : High School Math

Indicate the sum of the following series:

Explanation:

The formula for the sum of a geometric series is

,

where  is the first term in the series,  is the rate of change between sequential terms, and  is the number of terms in the series

For this problem, these values are:

Plugging in our values, we get:

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