Sum of arithmetic series:
formula and worked example

What a series sum is for

An arithmetic series is what you get when you add up a sequence whose consecutive terms differ by the same fixed amount. Monthly savings that go up by the same sum every month, seat counts in a stadium where each row has two more seats than the one in front, and steps in a staircase of equal height all form arithmetic patterns.

Adding such lists one term at a time is slow and error-prone. The shortcut rests on a symmetry: the first term plus the last term equals the second plus the second-last, and so on. Every pair from the two ends gives the same total, so you only need one pair and a count.

The same idea is behind the familiar trick of adding 1 to 100 in seconds. Pair 1 with 100, 2 with 99, and so on: fifty pairs of 101 give 5,050.

The formula and its three inputs

Sum equals the number of terms, times the first plus the last, divided by two.

• First: the opening term of the series, 4 in the example.
• Last: the closing term, 40 in the example.
• Terms: how many numbers are being added, counting both ends, 10 in the example.

The common difference is not an input, but it is hidden inside the other three. For the example it is (40 − 4) ÷ (10 − 1) = 4, so the list is 4, 8, 12 and so on up to 40.

Worked example: ten terms from 4 to 40

Add the sequence that starts at 4, steps by 4 and finishes at 40.

1. 1Add the first and last terms: 4 + 40 = 44.
2. 2Multiply by the number of terms: 10 × 44 = 440.
3. 3Halve it: 440 ÷ 2 = 220.

To confirm, add the list by hand: 4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40. The five end-to-end pairs each make 44, and five times 44 is 220.

Making sense of the total

The total is best understood as the average term multiplied by how many there are. Here the average term is 22, which is the midpoint of 4 and 40, and ten of those make 220. Any arithmetic series has its average at the midpoint of its first and last terms, which gives you a fast estimate by eye.

That reading is helpful for budgeting. If an instalment plan starts at 4,000 and rises by 4,000 each month for ten months, the total outlay is 220,000, not ten times either end. The average instalment is what matters, and it sits halfway between the smallest and the largest.

A series that falls works the same way. A declining balance that reduces by equal steps still has its average at the midpoint, so the same formula applies with the larger number as the first term.

How the first term moves the total

The sum is linear in the first term. Each unit added to the first term adds half the term count, here 5, to the total, because the first term appears once in every pair and there are five pairs.

A 10% rise in the first term only lifts the total by about 0.9% here, because the first term is small compared to the last. Changing the last term would shift the sum far more.

Mistakes that creep into series sums

• Using the wrong term count, for example 9 instead of 10 when both end values are included. Count the terms, or compute it as (last − first) ÷ difference + 1.
• Skipping a term inside the list. The first and last values may be unchanged, but the equal-step pattern is broken and the rule no longer applies.
• Applying this to a sequence whose steps are not equal, such as one that doubles each time. That needs the geometric series rule.
• Forgetting to halve, and reporting the sum of the pairs as the total.

Applying it to a payment ladder

A stall owner pays a rising daily fee: 50 on day one, increasing by 10 each day, for 14 days. Is the arithmetic rule the right tool, and what does it give?

1. 1Find the last fee: 50 + 13 × 10 = 180.
2. 2Check the term count: 14 days means 14 terms.
3. 3Sum: 14 × (50 + 180) ÷ 2 = 14 × 115 = 1,610.

Because the steps are equal, the rule fits. If the fee instead doubled each day, the series would be geometric and the answer would be much larger.