Introduction to Arithmetic Progressions

Recall that an ordered list of numbers, separated by commas, is called a progression or sequence, and when a pattern is detectable in a progression, a generating rule can often be established that enables us to find any term in the sequence. 

Arithmetic progressions start with a first term and then either increase or decrease by a constant amount called the common difference. We denote the first term by the letter $a$a and the common difference by the letter $d$d.

The progression $-3,5,13,21,\ldots$−3,5,13,21,… is an arithmetic progression with $a=-3$a=−3 and $d=8$d=8. On the other hand, the progression $1,10,100,1000,\ldots$1,10,100,1000,… is not arithmetic because the difference between each term is not constant.

Consider the arithmetic sequence $-4,3,10,17,...$−4,3,10,17,... We know that it is arithmetic because $3-\left(-4\right)=10-3=7$3−(−4)=10−3=7 and so with the common difference of $7$7, the next three terms are $24,31$24,31, and $38$38.

We also can develop a rule that generates an arithmetic progression. For example, take the rule given by  $t_n=80-7n$tn​=80−7n. We can use this formula to begin writing down the sequence. The first term, or $t_1$t1​ is found by substituting $n=1$n=1 into the formula, so $t_1=80-7\times1=73$t1​=80−7×1=73. Similarly $t_2=80-7\times2=66$t2​=80−7×2=66. We can find any term this way.

For example, the first negative term in the sequence is given by $t_{12}=80-7\times12=-4$t12​=80−7×12=−4.

This applet will allow you to explore the geometrical features of an arithmetic progression, change the values of $a$a (the initial term) and $d$d (the common difference). What shape is the line?

Practice questions

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QUESTION 3