Standard sum formulas cover the most common arithmetic series. The sum of the first \(n\) positive integers is \(S_n = \frac{n(n+1)}{2}\), so \(1 + 2 + \cdots + 100 = 5050\). The sum of the first \(n\) squares is \(1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}\). The sum of the first \(n\) cubes has the elegant closed form \(1^3 + 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2\), which is precisely the square of the integer sum. Finally, the sum of the first \(n\) odd numbers equals \(n^2\), so the first 50 odd numbers sum to 2500.
These formulas extend naturally to multiples and partial ranges. The sum of the first \(n\) even numbers is \(2 \cdot \frac{n(n+1)}{2} = n(n+1)\), giving 20 · 21 = 420 for the first 20 even numbers and 30 · 31 = 930 for the first 30. The sum of the first 100 multiples of 7 is \(7 \cdot \frac{100 \cdot 101}{2} = 7 \cdot 5050 = 35{,}350\). The sum of multiples of 3 from 3 to 99 has 33 terms averaging 51, giving 33 · 51 = 1683. For an arbitrary arithmetic series from \(a\) to \(b\) inclusive, the number of terms is \(b - a + 1\), the average is \((a+b)/2\), and the sum is the average times the term count. For example, \(11 + 12 + \cdots + 20\) has 10 terms averaging 15.5, summing to 155. The same averaging trick handles the median of consecutive integers: the average from \(a\) to \(b\) is always \((a+b)/2\), equal to the median.
Three classical means provide compact summaries of data. The arithmetic mean of \(v_1, \ldots, v_n\) is \(\frac{v_1 + \cdots + v_n}{n}\), so the arithmetic mean of 12, 18, 24, and 30 is \(84/4 = 21\). The geometric mean of two positive numbers \(a\) and \(b\) is \(\sqrt{ab}\); applied to 4 and 36 this gives \(\sqrt{144} = 12\). The harmonic mean is \(\frac{2ab}{a+b}\); for 2 and 8 it equals \(16/5 = 3.2\). For two equal distances traveled at speeds \(a\) and \(b\), the average round-trip speed is exactly the harmonic mean \(\frac{2ab}{a+b)}\). The median of an odd-sized ordered list is its middle value (so 3, 8, 11, 17, 25 has median 11), the mode is the most frequent value (9 in 2, 5, 5, 8, 9, 9, 9, 10), and the range is the maximum minus the minimum (15 for −3, 7, 12, 0, 5).