Arithmetic Series - Algebra 2

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 .

A simple way to find the term of an arithmetic sequence is to use the formula .

Here, is the term you are trying to find, is the first term, and is the common difference. For this question, the common difference is .

The common difference between the terms is half that between the second and fourth terms - that is:

Subtract this common difference from the second term to get the first:

You can either calculate the vaules of and and subtract, or notice from the formula that each succesive number in the sequence is 3 larger than the previous

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.

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.

In each case, the terms increase by the same number, so all of these sequences are arithmetic.

Each term is the result of adding 1 to the previous term. 1 is the common difference.

Each term is the result of subtracting 1 from - or, equivalently, adding to - the previous term. is the common difference.

The common difference is 0 in a constant sequence such as this.

Each term is the result of adding to the previous term. is the common difference.

First, find a pattern in the sequence. You will notice that each time you move from one number to the very next one, it increases by 7. That is, the difference between one number and the next is 7. Therefore, we can add 7 to 36 and the result will be 43. Thus .

Determine what kind of sequence you have, i.e. whether the sequence changes by a constant difference or a constant ratio. You can test this by looking at pairs of numbers, but this sequence has a constant difference (arithmetic sequence).

So the sequence advances by subtracting 16 each time. Apply this to the last given term.

This is a sequence that features every other positive, odd integers. The missing number in this case is 9.

In an arithmetic sequence the amount that the sequence grows or shrinks by on each successive term is the common difference. This is a fixed number you can get by subtracting the first term from the second.

So the sequence is adding 12 each time. Add 12 to 25 to get the third term.

So the unknown term is 37. To double check add 12 again to 37 and it should equal the fourth term, 49, which it does.

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:

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 .

The recursive formula for an arithmetic sequence is .

The only answer that fits this description is

.

Find the next term in the following arithmetic series:

To find the next term in an arithmetic series, we need to find the common difference. To do so, find the difference between any two consecutive terms in the sequence:

Our common difference is 7. Now we need to add that to the last term to get what we want

So our next term is 32

What is the common difference of the following arithmetic series?

To find the common difference, we need to find the difference between any two consecutive terms.

Try with the first two:

To be sure, try it with the 2nd and 3rd

We keep getting the same thing, -8. It must be negative, because our sequence is decreasing. Therefore, we have our answer: -8

The sequence is decreasing by 3 each term. To get from the first term to the 16th term, you must subtract 3 fifteen times:

Write the n-th term formula.

The represents the first term, and is the last term.

The is the common difference among the numbers.

since each term increases by two.

Solve for .

Divide by two on both sides.

The formula for n-terms in a arithmetic sequence is:

Substitute the known terms.

The answer is:

Write the formula for the sum of an arithmetic series.

To determine the value of , use the formula:

Divide by five on both sides.

Substitute all the terms into the sum formula.

The answer is: