Arithmetic Series Sum
Sum of the first n terms of an arithmetic sequence.
Formula
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Example
Sum 1..100 = 100/2 × (1+100) = 5050
How it works
This formula gives the total of an arithmetic sequence, where each term changes by a constant amount. It works because if you pair the first and last terms, the second and second-to-last terms, and so on, each pair has the same sum, so the whole list can be counted as equal pairs. That is why the total is half the number of terms times the sum of the first and last term.
When to use it
Use this when you need the sum of a list of numbers with a constant difference, such as savings that increase by the same amount each week or a pattern like 3, 7, 11, 15, ... . It is especially useful when you know the first and last terms but do not want to add every term one by one.
Worked example, step by step
Find the sum of the arithmetic sequence 4, 9, 14, 19, 24. Step 1: Identify the values. n = 5 a_1 = 4 a_n = 24 Step 2: Substitute into the formula. S_n = n/2 (a_1 + a_n) S_5 = 5/2 (4 + 24) Step 3: Add inside the parentheses. S_5 = 5/2 (28) Step 4: Multiply. S_5 = 70 Final answer: 70
Common mistakes to avoid
- Using the formula with a sequence that does not have a constant difference
- Forgetting that n is the number of terms, not the last term number
- Adding the terms incorrectly by using the average of all terms instead of the first and last term average
Did You Know?
Legend says a young Gauss summed 1 to 100 in seconds by pairing terms — this is the formula he effectively rediscovered.
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