Sequence Calculator

1. Choose the sequence type: arithmetic or geometric.
2. Enter the first term and the common difference or ratio.
3. Enter the term number n you want.
4. Read the nth term, the sum of the first n terms, and a preview.

Here is what each result means:

A sequence is an ordered list of numbers that follow a rule. In an arithmetic sequence, each term is the previous one plus a fixed amount, the common difference. In a geometric sequence, each term is the previous one times a fixed factor, the common ratio. So 2, 5, 8, 11 is arithmetic with a difference of 3, and 3, 6, 12, 24 is geometric with a ratio of 2.

These two patterns are the most common in algebra. Arithmetic sequences model steady change, like a fixed weekly saving. Geometric sequences model growth or decay by a percentage, like compound interest or a halving dose. Their formulas let you jump to any term or add many terms without listing them all.

It applies the nth-term and sum formulas for the type you choose.

1. For the nth term, apply the difference or ratio n minus 1 times.
2. For the sum, use the arithmetic or geometric series formula.
3. Preview the first few terms to check the pattern.

A geometric ratio is really repeated multiplication, so the exponent calculator handles the powers behind it.

Find the 10th term of the arithmetic sequence starting at 2 with difference 3.

Calculation: a_10 = 2 + (10 - 1) x 3 = 2 + 27 = 29. The sum of the first ten terms is (10/2)(2 x 2 + 9 x 3) = 5 x 31 = 155. So the 10th term is 29 and the running total is 155.

The two types differ in how each term is built from the last.

Arithmetic sequences change by the same amount each step; geometric sequences change by the same percentage, so they grow or shrink much faster.

Adding many terms uses a series formula rather than listing them.

A geometric sum with a ratio between minus one and one settles toward a fixed total as n grows, an infinite series that converges.

The Common Difference or Ratio

A larger step makes terms grow faster. A negative difference falls, and a ratio below one shrinks the terms.

The First Term

It scales the whole sequence, since every term builds from it.

The Number of Terms

More terms mean a larger sum, and for geometric growth the sum can rise very steeply.

Algebra Homework

Find a distant term or the sum without writing out every step.

Savings and Growth

Model steady saving as arithmetic, or percentage growth as geometric.

Checking a Pattern

Preview the first terms to confirm a rule fits your numbers.

1. Using N Instead of N Minus 1

The nth term applies the step n minus 1 times, since the first term takes no step.

2. Mixing Up Difference and Ratio

Arithmetic adds a difference; geometric multiplies by a ratio. Pick the right type first.

3. Wrong Sum Formula

Each type has its own series formula. Using the arithmetic sum for a geometric sequence is a common slip.

4. Forgetting a Negative Step

A negative difference or a ratio below one makes terms fall. That is valid, not an error.

5. Off-by-one on the Count

Be clear whether n counts the term you want or the number of terms to sum.

The formulas are exact; only the displayed decimals and very large geometric terms are rounded.

It is a list where each term is the previous one plus a fixed amount, the common difference. So 2, 5, 8, 11 is arithmetic with a difference of 3. Its nth term is a plus n minus 1 times the difference.

It is a list where each term is the previous one times a fixed factor, the common ratio. So 3, 6, 12, 24 is geometric with a ratio of 2. Its nth term is a times the ratio to the power n minus 1.

For arithmetic, add the common difference n minus 1 times to the first term. For geometric, multiply the first term by the ratio to the power n minus 1. This tool does both.

Arithmetic uses n over 2 times twice the first term plus n minus 1 times the difference. Geometric uses the first term times the ratio to the n, minus 1, over the ratio minus 1.

Arithmetic sequences add the same amount each step, giving straight-line change. Geometric sequences multiply by the same factor, giving faster growth or decay by a percentage.

Yes. A ratio between zero and one makes a geometric sequence shrink, like 100, 50, 25. A negative ratio makes the terms alternate in sign.

When the ratio is between minus one and one, the terms shrink toward zero and their infinite sum settles on a fixed total. Larger ratios make the sum grow without limit.

Because the first term takes no step. To reach the nth term you apply the difference or ratio n minus 1 times, starting from the first term.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Arithmetic progression (Wikipedia).
• Sequences and series (Maths Is Fun).
• Geometric progression (Wikipedia).