Calculate the Sum of Terms of an Arithmetic Sequence (the Series Sum / or Gauss), Given the First Term of the Series, the Common Difference and the Last Term. Integers, Decimal Numbers and Fractions Are Accepted as Input Values. Online Calculator: Learn How To Add All the Terms, Formula, Explanations and the Result

Calculate the Sum S_n of the Terms of the Arithmetic Sequence (Series)

1. If a = the first term of the series, a_n = the last term of the series, d = the common difference, n = the number of terms of the series, and if a, a_n and d are being known, then from the formula: a_n = a + (n – 1) × d we calculate n. 2. Substitute for the values we know in the formula of the Sum to calculate it: S_n = ⁿ/₂ × (a + a_n)

The Latest Calculated Sums of Arithmetic Series Terms, Given the First Term, the Common Difference and the Last Term

Calculate Sum of Arithmetic Series Terms, Given the First Term, -11/2, the Common Difference, -2 and the Last Term, -27/2	Apr 04 00:16 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -11/2, the Common Difference, -2 and the Last Term, -27/2	Mar 31 22:34 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -2, the Common Difference, -1/2 and the Last Term, -85.5	Mar 31 22:34 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20.5	Mar 31 22:34 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, 1, the Common Difference, 1 and the Last Term, 204	Mar 11 08:13 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20.5	Mar 11 08:13 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20	Mar 11 08:13 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, 1, the Common Difference, 1 and the Last Term, 204	Mar 01 13:54 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20	Mar 01 13:54 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20.5	Mar 01 13:54 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20	Feb 05 23:12 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, -1/2, the Common Difference, -1/2 and the Last Term, -20.5	Feb 05 23:12 UTC (GMT)
Calculate Sum of Arithmetic Series Terms, Given the First Term, 1, the Common Difference, 1 and the Last Term, 204	Feb 05 23:12 UTC (GMT)
All the Calculated Sums of Arithmetic Series Terms, Given the First Term, the Common Difference and the Last Term

Sums of Arithmetic Series (also called Arithmetic Sequences, or Arithmetic Progressions)

What is an Arithmetic Sequence (also called an Arithmetic Series or an Arithmetic Progression)?

An arithmetic sequence is an ordered set of numbers in which the difference between any consecutive terms is the same. This difference is called the common difference of the arithmetic sequence. The ordered numbers are called the terms of the sequence.

Examples of a arithmetic sequences:
2, 4, 6, 8, 10. The common difference is 4 - 2 = 6 - 4 = 8 - 6 = 10 - 8 = 2.
1, 5, 9, 13, 17, 21, 25. The common difference is 5 - 1 = 9 - 5 = 13 - 9 = 17 - 13 = 21 - 17 = 25 - 21 = 4.

The difference between an arithmetic sequence and an arithmetic series.

When the terms of the ordered set of numbers are written as a comma separated string, we call this an arithmetic sequence.

When the same terms are written as a summation, then we call this an arithmetic series.

What does an arithmetic sequence look like, in general?

If we call a = the first term of an arithmetic sequence, d = the common difference, n = the number of terms in the sequence, then the terms of the arithmetic sequence will be:
The first term: a
The second term:
a₂ = a + d
The third term:
a₃ = a₂ + d = a + d + d = a + 2 × d
The fourth term:
a₄ = a₃ + d = a + 2 × d + d = a + 3 × d
...
The k^th term:
a_k = a_k-1 + d = a + (k - 2) × d + d = a + (k - 1) × d
The n^th term, the last one: a_n = a_n-1 + d = a + (n - 2) × d + d = a + (n - 1) × d

How to get to the formula for the Sum S_n of an arithmetic series terms?

We write the arithmetic series in two ways.
First we write it from the term a to the term a_n, then we write it "inverted", from the term a_n to a:
S_n = a + a₂ + a₃ + ... + a_(n-2) + a_(n-1) + a_n
S_n = a_n + a_(n-1) + a_(n-2) + ... + a₃ + a₂ + a
Then we add up these two identical series S_n, that were written differently:
S_n + S_n = (a + a_n) + (a₂ + a_(n-1)) + (a₃ + a_(n-2)) + ... + (a_(n-2) + a₃) + (a_(n-1) + a₂) + (a_n + a) =>
2 × S_n = (a + a + (n - 1) × d) + (a + d + a + (n - 2) × d) + (a + 2 × d + a + (n - 3) × d) + ... + (a + (n - 3) × d + a + 2 × d) + (a + (n - 2) × d + a + d) + (a + (n - 1) × d + a) =>
2 × S_n = (2 × a + (n - 1) × d) + (2 × a + (n - 1) × d) + (2 × a + (n - 1) × d) + ... + (2 × a + (n - 1) × d) + (2 × a + (n - 1) × d) + (2 × a + (n - 1) × d) =>
2 × S_n = n × (2 × a + (n - 1) × d) =>
S_n = ⁿ/₂ × (2 × a + (n - 1) × d)

The formula for the sum of an arithmetic series:

S_n = ⁿ/₂ × (2 × a + (n - 1) × d)
n = the number of terms in the arithmetic series
a = the first term of the arithmetic series
d = the common difference of the arithmetic series
a_n = the last term of the arithmetic series
...
Furthermore, knowing that a_n = a + (n - 1) × d
We can also write the sum S_n in the form:
S_n = ⁿ/₂ × (a + a_n)

Given the First Term, the Common Difference and the Last Term, How to Calculate the Sum of the Terms of the Arithmetic Series?

We use the formula of the last term, a_n:
a_n = a + (n - 1) × d =>
a_n - a = (n - 1) × d =>
(a_n - a) ÷ d = (n - 1) =>
(a_n - a) ÷ d + 1 = n =>
n = (a_n - a) ÷ d + 1

Calculate the Sum Sn of the Terms of the Arithmetic Sequence (Series)