Sequences and Series

Lesson

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 $t_n=80-7n$`t``n`=80−7`n`. We can use this formula to begin writing down the sequence. The first term, or $t_1$`t`1 is found by substituting $n=1$`n`=1 into the formula, so $t_1=80-7\times1=73$`t`1=80−7×1=73. Similarly $t_2=80-7\times2=66$`t`2=80−7×2=66. We can find any term this way.

For example, the first negative term in the sequence is given by $t_{12}=80-7\times12=-4$`t`12=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?

Study the pattern for the following sequence, and write down the next two terms.

$6$6, $2$2, $-2$−2, $-6$−6, $\editable{}$, $\editable{}$

Study the pattern for the following sequence.

$280$280, $230$230, $180$180, $130$130 ...

State the common difference between consecutive terms.

A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after $1$1 minute, $2$2 minutes and $3$3 minutes is $50$50 metres, $100$100 metres and $150$150 metres respectively.

By how much is the depth increasing each minute?

What will the depth of the vessel be after $4$4 minutes?

Continuing at this rate, what will be the depth of the vessel after $10$10 minutes?