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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.
The simplest type of numerical sequence is an 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 |
$\vdots $ | |
${n}^{\mathrm{th}}$ term: | $n=(n-1)+1$ |
Find the constant value that is added to get the following sequences and write out the next 5 terms.
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:
Thus, the equation for the ${n}^{th}$ -term will be:
Given ${a}_{1}$ and the common difference, $d$ , the entire set of numbers belonging to an arithmetic sequence can be generated.
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:
where
A simple test for an arithmetic sequence is to check that the difference between consecutive terms is constant:
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 the terms of sequence sometimes helps in determining the type of sequence involved.For an arithmetic sequence, plotting ${a}_{n}$ vs. $n$ results in:
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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