A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one is not constant:

These ratios can be immediately seen to be unequal as they are of different sign.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term is the same:

The sequence could be arithmetic.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one is the same:

The sequence could be geometric.

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.

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

Write the formula for the arithmetic sequence.

The first term is:

The common difference is the same for each term, which is increasing by six every term:

Substitute and simplify the formula.

To find the hundredth term, plug in .

The answer is:

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

a) The general formula for an arithmetic sequence of terms is written,

is the first term in the sequence, and is the common difference, which is just the difference between any two adjacent terms in the sequence.

For the sequence to be a true arithmetic sequence, the common difference must be the same for any two adjacent terms in the sequence.

For our sequence:

and so on...

An explicit formula can now be written,

We can now test this formula for the first few terms (columns 1-3 in the table below).

b) In the table below, columns 4-5 show the calculations for the term and the term. The difference is simply .

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

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: