Arithmetic Series Simplified Revision Notes for Leaving Cert Mathematics

Revision notes with simplified explanations to understand Arithmetic Series quickly and effectively.

Learn about Arithmetic Sequences & Series for your Leaving Cert Mathematics Exam. This Revision Note includes a summary of Arithmetic Sequences & Series for easy recall in your Mathematics exam

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Arithmetic Series

Introduction

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A series is a sequence summed up.

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An arithmetic sequence is an arithmetic sequence summed up.

The general term of any arithmetic series is

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General term of an arithmetic series : Page 22

S_n=\tfrac{n}{2}[2a+(n-1)d]

Where $S_n$ is the sum of the first $n$ terms in an arithmetic sequence :

S_n=T_1+T_2+...+T_n

Example

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Find the sum of the first 25 terms in an arithmetic sequence formed by $T_n=2n-6$

First we need to determine a and d.

METHOD 1 Find first few terms and identify a and d :

\begin{align*} T_1&=2(1)-6=-4=a \\ T_2&=2(2)-6=-2 \\ T_3&=2(3)-6=0 \\ \end{align*}

From observation we can see that the common difference is 2.

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METHOD 2 We can also identify a and d by equating terms.

Rearrange the general term of an arithmetic sequence :

\begin{align*} T_n&=a+(n-1)d \\ &=a+nd-d \\ &=dn+(a-d) \end{align*}

Now equate this equation and the general the we're given :

\begin{align*} dn+(a-d)&=2n-6 \\ \end{align*}

$n$ term : $d=2$
constant term : $(a-(2))=-6$ Solving for a gives us -4.

Now let's derive the sum of the first 25th term :

\begin{align*} S_n&=\tfrac{n}{2}[2a+(n-1)d] \\ S_{25}&=\tfrac{25}{2}[2(-4)+(25-1)(2)] \\ &=:highlight[500] \end{align*}

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