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Algebra / Sequences and Series

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. There's a famous shortcut for it, the one a young Carl Friedrich Gauss is said to have used to add 1 to 100 in seconds, and it works for any arithmetic sequence you'll meet.

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Chapter 8 · Lesson 6 of 8Sequences and Series
  1. 1.Arithmetic Sequences
  2. 2.Geometric Sequences
  3. 3.Arithmetic vs Geometric Sequences
  4. 4.Recursive Formulas
  5. 5.Sigma Notation
  6. 6.Arithmetic Series
  7. 7.Geometric Series
  8. 8.Binomial Theorem
See all algebra chapters →

The story goes that a schoolteacher, wanting a quiet half-hour, told his class to add up every whole number from 1 to 100. Young Carl Friedrich Gauss came back with 5050 almost immediately. He’d noticed that 1 + 100, 2 + 99 and 3 + 98 all make 101, and that there are 50 such pairs. That pairing trick is the whole idea behind the arithmetic series formula, the sum of the terms of an arithmetic sequence.

Arithmetic series formula: S sub n equals n times (a sub 1 plus a sub n) divided by 2
The sum of an arithmetic series

The Arithmetic Series Formula

The sum of the first terms of an arithmetic sequence is

In words: the number of terms times the average of the first and last terms. If you don’t know the last term, substitute to get the second version:

SymbolMeaning
the sum of the first terms
the number of terms
, the first and last terms
the common difference

In sigma notation, the same sum is .

Why the Formula Works

Gauss’s trick pairs the first term with the last, the second with the second-to-last, and so on. In an arithmetic sequence every pair has the same sum, , because as one end goes up by , the other goes down by .

The numbers 1 to 100 paired from opposite ends: 1 with 100, 2 with 99, 3 with 98, through 50 with 51; each pair sums to 101, and 50 pairs times 101 equals 5050
Every pair adds to the same total, first plus last

There are pairs, each worth , so the total is . If is odd, the pairing leaves a middle term over, but the formula still works. Here’s a picture that avoids the odd/even question altogether: put two copies of the sum together, one flipped, and they form a rectangle.

A staircase of 1 + 2 + 3 + 4 + 5 + 6 blue squares and the same staircase flipped in green fit together into a 6 by 7 rectangle of 42 squares, so one staircase is 21
Two copies of the sum make a rectangle, so one copy is half of it

How to Find the Sum of an Arithmetic Series

  1. Find , and . If you’re given the last term instead of , solve for .
  2. Find the last term if you need it.
  3. Apply the formula.

Example 1: The First 20 Terms

Find the sum of the first 20 terms of 3, 7, 11, …

Step 1 — the pieces: , , .

Step 2 — the last term:

Step 3 — the sum:

Example 2: When You Know the Last Term

Find .

Step 1 — count the terms:

Step 2 — the sum:

Example 3: Using the Second Formula

Find the sum of the first 15 terms of 4, 9, 14, …, without finding the last term first.

Finding How Many Terms Add Up to a Total

How many terms of add up to 110?

Step 1 — write the sum with , :

Step 2 — solve the quadratic:

The negative root doesn’t make sense as a number of terms, so 10 terms.

Arithmetic Series Word Problems

Seats in a Theater

The front row has 20 seats and each row has 2 more than the one in front, for 15 rows. How many seats are there in total?

Step 1 — the last row: .

Step 2 — the total:

A Savings Plan

You save the first month and more each month after that. How much have you saved after a year?

Step 1 — the 12th month: .

Step 2 — the total:

You’ve saved .

Useful Special Cases

SumFormulaExample
(first odd numbers)first 30 odd numbers:
(first even numbers)first 10 even numbers:

Common Mistakes to Avoid

  • Miscounting the terms. From 5 to 62 in steps of 3 is 20 terms, not 19 or 57. Use the sequence formula to count.
  • Using instead of . The sum is times the average of the ends, so divide by 2.
  • Adding the wrong last term. Find before using .
  • Using the arithmetic formula on a geometric sequence. Check that the differences are constant first.
  • Keeping a negative . When a quadratic gives two values, only a positive whole number counts.

Practice Problems

Work each one before opening the answer.

Problem 1. Find .

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Answer:

Problem 2. Find the sum of the first 15 terms of 4, 9, 14, …

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Step 1 — the last term: .

Step 2 — the sum:

Answer:

Problem 3. Find .

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Step 1 — 20 terms, from 10 to 200 in steps of 10.

Step 2 — the sum:

Answer:

Problem 4. Find the sum of the first 30 odd numbers.

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Step 1 — , , so .

Step 2 — the sum:

Answer:

Problem 5. Evaluate .

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Step 1 — the first and last terms: and , with 12 terms.

Step 2 — the sum:

Answer:

Problem 6. Logs are stacked with 20 on the bottom row, 19 on the next, and so on up to 1 on top. How many logs are there?

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Answer: logs

Adding a fixed amount each time gives a total that grows steadily. When each term is a fixed multiple of the last, the sum behaves very differently, and can even settle on a finite total after infinitely many terms. That’s a geometric series.

Frequently Asked Questions

What is the formula for an arithmetic series?+

The sum of the first n terms is : the number of terms times the average of the first and last terms. If you don't know the last term, use .

What is the difference between an arithmetic sequence and an arithmetic series?+

A sequence is the list of terms, like 3, 7, 11, 15. A series is the sum of those terms, 3 + 7 + 11 + 15 = 36. The sequence formula finds one term; the series formula finds a total.

How do you find the sum of 1 to 100?+

Pair the first and last numbers: 1 + 100, 2 + 99, 3 + 98, and so on. Each pair adds to 101, and there are 50 pairs, so the sum is 50 × 101 = 5050. The formula gives the same: 100 × (1 + 100) ÷ 2.

How do you find the number of terms in an arithmetic series?+

Use the sequence formula on the last term: , and solve for n. For 5 + 8 + 11 + … + 62, gives n = 20.

Can an arithmetic series be infinite?+

An infinite arithmetic series never settles on a finite sum unless every term is zero, because the terms keep adding a fixed amount. Infinite sums that do converge come from geometric series with a ratio between −1 and 1.

Related lessons

Arithmetic SequencesSigma NotationGeometric SeriesArithmetic vs Geometric Sequences

Calculators for this topic

Arithmetic Sequence CalculatorSummation Calculator