AP Sum When the Last Term Is Known

Quick answer: If an AP has n terms, first term a and last term l, its sum is Sₙ = n(a + l)/2. Each end-pair has the same sum; for odd n the middle term is half that pair sum.

Worked example

Find the sum 5 + 8 + 11 + … + 32.

1. Use l = a + (n − 1)d: 32 = 5 + 3(n − 1).
2. Thus n − 1 = 9 and n = 10.
3. S₁₀ = 10(5 + 32)/2 = 185.

Check the result

Five end-pairs each total 37, so their combined sum is 5 × 37 = 185.

Common mistake to avoid

The number of gaps is n − 1, not n. Do not apply this formula to a sequence without constant differences.

Try it yourself

1. Sum the first 20 positive integers. 2. Sum 4 + 9 + 14 + … + 49.

Answers and reasoning

1. 20(1 + 20)/2 = 210. 2. There are 10 terms; sum = 10(4 + 49)/2 = 265.

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