# Arithmetic & Geometric sequences, recursive formulae

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## Introduction

In this chapter we extend the arithmetic and quadratic sequences studied in earlier grades, to geometric sequences. We also look at series, which is the summing of the terms in a sequence.

## Arithmetic sequences

The simplest type of numerical sequence is an arithmetic sequence .

Arithmetic Sequence

An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term

For example, $1,2,3,4,5,6,...$ is an arithmetic sequence because you add 1 to the current term to get the next term:

 first term: 1 second term: 2=1+1 third term: 3=2+1 $⋮$ ${n}^{\mathrm{th}}$ term: $n=\left(n-1\right)+1$

## Common difference :

Find the constant value that is added to get the following sequences and write out the next 5 terms.

1. $2,6,10,14,18,22,...$
2. $-5,-3,-1,1,3,...$
3. $1,4,7,10,13,16,...$
4. $-1,10,21,32,43,54,...$
5. $3,0,-3,-6,-9,-12,...$

## General equation for the ${n}^{th}$ -term of an arithmetic sequence

More formally, the number we start out with is called ${a}_{1}$ (the first term), and the difference between each successive term is denoted d , called the common difference .

The general arithmetic sequence looks like:

$\begin{array}{ccc}\hfill {a}_{1}& =& {a}_{1}\hfill \\ \hfill {a}_{2}& =& {a}_{1}+d\hfill \\ \hfill {a}_{3}& =& {a}_{2}+d=\left({a}_{1}+d\right)+d={a}_{1}+2d\hfill \\ \hfill {a}_{4}& =& {a}_{3}+d=\left({a}_{1}+2d\right)+d={a}_{1}+3d\hfill \\ \hfill ...\\ \hfill {a}_{n}& =& {a}_{1}+d·\left(n-1\right)\hfill \end{array}$

Thus, the equation for the ${n}^{th}$ -term will be:

${a}_{n}={a}_{1}+d·\left(n-1\right)$

Given ${a}_{1}$ and the common difference, $d$ , the entire set of numbers belonging to an arithmetic sequence can be generated.

Arithmetic Sequence

An arithmetic (or linear ) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term:

$\begin{array}{c}\hfill {a}_{n}={a}_{n-1}+d\end{array}$

where

• ${a}_{n}$ represents the new term, the ${n}^{th}$ -term, that is calculated;
• ${a}_{n-1}$ represents the previous term, the ${\left(n-1\right)}^{th}$ -term;
• $d$ represents some constant.
Test for Arithmetic Sequences

A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant:

${a}_{2}-{a}_{1}={a}_{3}-{a}_{2}={a}_{n}-{a}_{n-1}=d$

This is quite an important equation, and is the definitive test for an arithmetic sequence. If this condition does not hold, the sequence is not an arithmetic sequence.

## Plotting a graph of terms in an arithmetic sequence

Plotting a graph of the terms of sequence sometimes helps in determining the type of sequence involved.For an arithmetic sequence, plotting ${a}_{n}$ vs. $n$ results in:

## Geometric sequences

Geometric Sequences

A geometric sequence is a sequence in which every number in the sequence is equal to the previous number in the sequence, multiplied by a constant number.

This means that the ratio between consecutive numbers in the geometric sequence is a constant. We will explain what we mean by ratio after looking at the following example.

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