AI Tutor
How did a 10-year-old child add every number from 11 to 100100 in under ten seconds without a calculator?
In the late 1700s, young Carl Friedrich Gauss realized that instead of adding sequentially, you can pair numbers from opposite ends: 1+100=1011 + 100 = 101, 2+99=1012 + 99 = 101, and so on.
Can this simple pairing trick find the sum of any sequence that grows by a constant step?

The Arithmetic Series Formula

An arithmetic sequence is an ordered list where each term changes by adding a fixed common difference (dd).
An arithmetic series is the indicated sum of the terms of an arithmetic sequence: Sn=a1+a2+a3+⋯+an=∑k=1nakS_n = a_1 + a_2 + a_3 + \dots + a_n = \sum_{k=1}^{n} a_k.
Writing the series forward and backward gives 2Sn=n(a1+an)2S_n = n(a_1 + a_n), which gives us our main formula when divided by 2.
What happens if a problem gives you the first and last terms, but doesn't tell you how many terms exist?

Finding Missing Information First

Let's find the sum of the series 7+12+17+⋯+1427 + 12 + 17 + \dots + 142.
We know a1=7a_1 = 7, d=5d = 5, and an=142a_n = 142, but we need the number of terms (nn) before we can sum.