# 5050

5050 is close to 5150 which can be interpreted as the radio call for an involuntary psychiatric hold. But that’s not why we’re here.

5050 can also be expressed as a summation. A summation consists of an expression containing a dummy variable through which the dummy variable is used to perform some calculation. $\sum_{n=1}^{\infty} 2^{-n} = 1$

This summation specifically is an infinite summation. The summation can also be called an infinite series.

Before diving into summations we must first understand sequences and series. Sequences are ordered lists of numbers. There are no operations being performed on the numbers, they exist only as an ordered list. ${1, 1, 2, 3, 5, 8, ...}$

A series is the sum of a sequence. Where above we have the sequence, the series would be: ${1 + 1 + 2 +3 + 5 + 8 + ...}$

Finite summations and series become especially useful when they can be manipulated to fit an existing form for which there is a known formula. $\sum_{n=1}^{n} = \frac{n(n+1)}{2}$

Read plainly: The sum from n = 1 to n items is equal to n times the quantity n plus 1 all over 2. It’s said that this formula can be attributed to Gauss. Where Gauss was given the task of adding all integers 1 through 100 but through an observation was able to derive the above formula. $\sum_{n=1}^{100} = \frac{100(100 + 1)}{2} = 5050$

Consider the series that represents this summation: ${1 + 2 + 3 + ... + 98 + 99 + 100}$ The series can be split into two series consisting of 50 terms each and rewritten as follows. $\{1 + 2 + 3 + ... + 48 + 49 + 50\} \\ + \\ \{100 + 99 + 98 + ... + 53 + 52 + 51\}$

Adding these vertically the following series is obtained. $\{ 101 + 101 + 101 + ... + 101 + 101 + 101 \}$ Of notable interest is the series now contains 50 terms, all being the number 101. To complete the whole picture consider another way to write 50 in terms of a number divided by 2: $\frac{100}{2}$

When the value of all the terms, 101, is multiplied by the alternative way to write 50 the derived formula become apparent. $\frac{100(101)}{2} = \frac{100(100 + 1)}{2} = 50 \cdot 101 = 5050$