# Algebra II : Arithmetic Series

## Example Questions

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### Example Question #63 : Mathematical Relationships And Basic Graphs

List the first 4 terms of an arithmetic sequence with a first term of 3, and a common difference of 5.

Explanation:

An arithmetic sequence is one in which the common difference is added to one term to get the next term. Thus, if the first term is 3, we add 5 to get the second term, and continue in this manner.

Thus, the first four terms are:

### Example Question #64 : Mathematical Relationships And Basic Graphs

Consider the following sequence:

Find the th term of this sequence.

Explanation:

This is an arithmetic sequence since the difference between consecutive terms is the same ().  To find the th term of an arithmetic sequence, use the formula

,

where  is the first term,  is the number of terms, and  is the difference between terms.  In this case,  is is , and  is

### Example Question #1 : How To Find The Next Term In An Arithmetic Sequence

In the following arithmetic sequence, what is ?

6

2

7

5

6

Explanation:

The question states that the sequence is arithmetic, which means we find the next number in the sequence by adding (or subtracting) a constant term. We know two of the values, separated by one unknown value.

We know that  is equally far from -1 and from 13; therefore  is equal to half the distance between these two values. The distance between them can be found by adding the absolute values.

The constant in the sequence is 7. From there we can go forward or backward to find out that .

### Example Question #2 : How To Find The Next Term In An Arithmetic Sequence

Given the sequence below, what is the sum of the next three numbers in the sequence?

Explanation:

By taking the difference between two adjacent numbers in the sequence, we can see that the common difference increases by one each time.

Our next term will fit the equation , meaning that the next term must be .

After , the next term will be , meaning that the next term must be .

Finally, after , the next term will be , meaning that the next term must be

The question asks for the sum of the next three terms, so now we need to add them together.

### Example Question #1 : How To Find The Common Difference In Sequences

Which of the following cannot be three consecutive terms of an arithmetic sequence?

Explanation:

In each group of numbers, compare the difference of the second and first terms to that of the third and second terms. The group in which they are unequal is the correct choice.

The last group of numbers is the correct choice.

### Example Question #2 : How To Find The Common Difference In Sequences

Consider the arithmetic sequence

If , find the common difference between consecutive terms.

Explanation:

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence. When solving this equation, one approach involves substituting 5 for  to find the numbers that make up this sequence. For example,

so 14 is the first term of the sequence. However, a much easier approach involves only the last two terms,  and .

The difference between these expressions is 8, so this must be the common difference between consecutive terms in the sequence.

### Example Question #1 : How To Find Consecutive Integers

Write a rule for the following arithmetic sequence:

Explanation:

Know that the general rule for an arithmetic sequence is

,

where  represents the first number in the sequence,  is the common difference between consecutive numbers, and  is the -th number in the sequence.

In our problem, .

Each time we move up from one number to the next, the sequence increases by 3.  Therefore,

The rule for this sequence is therefore .

### Example Question #65 : Mathematical Relationships And Basic Graphs

Which of the following could be the recursive formula for an arithmetic sequence?

Explanation:

The recursive formula for an arithmetic sequence is .

The only answer that fits this description is

.

### Example Question #1 : Arithmetic Series

Find the 20th term in the following series:

Explanation:

This is an artithmetic series.  The explicit formula for an arithmetic sequence is:

Where represents the term, and is the common difference.

In this instance .  Therefore:

### Example Question #2 : Arithmetic Series

Find the 35th term in this series:

Explanation:

This is an arithmetic series. The formula to find the th term is:

where is the difference between each term.

To find the 35th term substitute for and

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