Arithmetic and Geometric Sequence Calculator: nth Term, Sum of n Terms and Sum to Infinity

Find terms and sums of arithmetic and geometric sequences.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose arithmetic or geometric and enter the first term and the common difference or ratio.

Enter n, and optionally a value to find the position of.

The calculator lists the first terms and gives the nth term, the sum and, for a geometric series with ratio between -1 and 1, the sum to infinity.

Formulas used

  • a_n = a + (n - 1)d
  • S_n = \frac{n}{2}\bigl(2a + (n - 1)d\bigr)
  • a_n = ar^{n-1}
  • S_n = \frac{a(1 - r^n)}{1 - r},\quad S_\infty = \frac{a}{1 - r}

Worked example

With the default numbers in the calculator above:

  1. The first terms: 3, 7, 11, 15, 19, 23, 27, 31, ...
  2. The nth term: 39.00
  3. Sum of the first n terms: 210.00
  4. Term number of the value you gave: 10.00 (it is term number 10)

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Sequence Geometric: multiply by the same amount each time; First term 2; Common difference d (arithmetic) or common ratio r (geometric) 0.5.

  1. The first terms: 2, 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, ...
  2. The nth term: 0.0039
  3. Sum of the first n terms: 4.00
  4. Sum to infinity (it exists because the ratio is between -1 and 1): 4.00
  5. Term number of the value you gave: -3.29 (not a whole number, so the value is not in the sequence)

Common questions

How do I tell arithmetic from geometric?

If the difference between terms is constant it is arithmetic. If the ratio of terms is constant it is geometric.

When does a geometric series have a sum to infinity?

Only when the ratio is between -1 and 1, so that the terms shrink to zero.

What is Gauss's trick?

Pair the first and last terms: each pair adds to the same total, so the sum is n/2 times (first + last).

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any school or first-year mathematics text (sequences and series).

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