Special Sums and Means

Power sums

Sum of a constant

\sum_{k = 1}^{n} c = c n

Sum of the first n numbers

\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}

Sum of odd numbers

\sum_{k = 1}^{n} (2 k - 1) = n^{2}

Sum of even numbers

\sum_{k = 1}^{n} 2 k = n (n + 1)

Sum of squares

\sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}

Sum of cubes

\sum_{k = 1}^{n} k^{3} = {(\frac{n (n + 1)}{2})}^{2}

Telescoping sum

\sum_{k = 1}^{n} \frac{1}{k (k + 1)} = 1 - \frac{1}{n + 1} = \frac{n}{n + 1}

nth term

a_{n} = \frac{1}{\frac{1}{a_{1}} + (n - 1) d}

the reciprocals form an arithmetic sequence with difference d

Harmonic mean

\frac{2 a b}{a + b} = \frac{2}{\frac{1}{a} + \frac{1}{b}}

Three terms in a row

b = \frac{2 a c}{a + c}

Inequality of means

AM \geq GM \geq HM

equal only when a = b

Link between the means

{GM}^{2} = AM \times HM

nth term

T_{n} = (a + (n - 1) d) r^{n - 1}

Sum of n terms

S_{n} = \frac{a}{1 - r} + \frac{d r (1 - r^{n - 1})}{(1 - r)^{2}} - \frac{(a + (n - 1) d) r^{n}}{1 - r}

r ≠ 1

Infinite sum

S_{\infty} = \frac{a}{1 - r} + \frac{d r}{(1 - r)^{2}}

|r| < 1

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