Arithmetic Sequence Calculator

An arithmetic sequence goes up or down by the same amount each time, such as 3, 7, 11, 15. That fixed amount is the common difference. This calculator finds any term and the sum of the first n terms using the standard formulas, and checks them by generating the terms one by one.

Try:

How to use the arithmetic sequence calculator

  1. Enter the first term a₁.
  2. Enter the common difference d and the term number n.
  3. Press Calculate to see aₙ, Sₙ and the first terms.

Formula

an=a1+(n−1)da_n = a_1 + (n-1)d
Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)

Worked example: 3, 7, 11, … (n = 10)

  • First term a₁: 3
  • Common difference d: 4
  • Term number n: 10

✓ Answer checked

a10=39,S10=210a_{10} = 39,\quad S_{10} = 210
First terms
3, 7, 11, 15, 19, 23, 27, 31, …
01020301: 312: 723: 1134: 1545: 1956: 2367: 2778: 3189: 35910: 3910
The terms go up by the same amount (4) each time, so the bars make a straight staircase.
Step-by-step working (2 steps)

  1. nth term formula aₙ = a₁ + (n − 1)d

    Start at a₁ and add d, (n − 1) times.

    an=a1+(n−1)d=3+(10−1)(4)=39a_n = a_1 + (n-1)d = 3 + (10-1)(4) = 39

  2. Sum formula Sₙ = n(a₁ + aₙ)/2

    Average of the first and last term, times the number of terms.

    Sn=n(a1+an)2=10(3+39)2=210S_n = \frac{n(a_1 + a_n)}{2} = \frac{10(3 + 39)}{2} = 210
Formulas used
an=a1+(n−1)da_n = a_1 + (n-1)d
Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)
How this was checked
  • ✓ Generating all 10 terms one by one gives the same last term and sum.

Frequently asked questions

How do I find the nth term?

Start with the first term and add the common difference (n − 1) times: aₙ = a₁ + (n − 1)d.

What is the sum of 1 to 100?

5050. There are 100 terms with average (1 + 100) ÷ 2 = 50.5, and 100 × 50.5 = 5050.

Can the common difference be negative?

Yes. Then the sequence decreases, like 100, 95, 90, ….

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