Finite Arithmetic Series

Finite Arithmetic Series

An arithmetic sequence is a sequence of numbers, such that the difference between any term and the previous term is a constant number called the common difference (\(d\)):

\[{T}_{n}={a} + (n-1)d\]

where

• \({T}_{n}\) is the \(n\)\(^{\text{th}}\) term of the sequence;

• \({a}\) is the first term;

• \(d\) is the common difference.

When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series.

The sum of the first one hundred integers

A simple arithmetic sequence is when \({a} = 1\) and \(d=1\), which is the sequence of positive integers:

\begin{align*} {T}_{n}& = {a} + (n-1)d \\ & = 1 + (n-1)(1) \\ & = n \\ \therefore \{{T}_{n}\} & = 1; 2; 3; 4; 5; \ldots \end{align*}

If we wish to sum this sequence from \(n=1\) to any positive integer, for example \(\text{100}\), we would write

\[\sum _{n=1}^{100}n=1+2+3+\cdots + 100\]

This gives the answer to the sum of the first \(\text{100}\) positive integers.

The mathematician, Karl Friedrich Gauss, discovered the following proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from \(\text{1}\) to \(\text{100}\). Young Karl quickly realised how to do this and shocked the teacher with the correct answer, \(\text{5 050}\). This is the method that he used:

• Write the numbers in ascending order.
• Write the numbers in descending order.
• Add the corresponding pairs of terms together.
• Simplify the equation by making \(S_{n}\) the subject of the equation.

\begin{align*} S_{100} &= \text{1} \enspace + \enspace \text{2} \enspace + \enspace \text{3} + \cdots + \text{98} + \text{99} + \text{100} \\ + \quad \underline{S_{100 }} &= \underline{ \text{100} + \text{99} +\text{98} + \cdots + \text{3} \enspace + \enspace \text{2} \enspace + \enspace \text{1}} \\ \therefore 2S_{100} &= \text{101} + \text{101} +\text{101} + \cdots + \text{101} +\text{101} +\text{101} \\ \therefore 2S_{\text{100}} &= \text{101} \times \text{100} \\ &= \text{10 100} \\ \therefore S_{\text{100}} &= \cfrac{\text{10 100}}{2} \\ &= \text{5 050} \end{align*}