Sequences and Series

Hong Kong

Stage 4 - Stage 5

Lesson

We have already seen that an arithmetic progression is defined a sequence of numbers where the difference between any two consecutive terms is a fixed constant $d$`d`. This means that each new term is generated by adding an amount $d$`d` to the previous term. Starting with the first term $t_1=a$`t`1=`a`, the second term is simply given by $t_2=t_1+d$`t`2=`t`1+`d` and the third term is given by $t_3=t_2+d$`t`3=`t`2+`d`, and the fourth term is given by $t_4=t_3+d$`t`4=`t`3+`d` and so on.

This means that the $\left(n+1\right)$(`n`+1)th term is given by $t_{n+1}=t_n+d$`t``n`+1=`t``n`+`d`. For example, the $100$100th term is given by $t_{100}=t_{99}+d$`t`100=`t`99+`d`.

We thus can define an arithmetic progression with what is known as a recursive formula. We begin by indicating a first term, say $t_1=a$`t`1=`a`, and then expressing the progression as:

$t_{n+1}=t_n+d,$`t``n`+1=`t``n`+`d`, with $t_1=a$`t`1=`a`

So for example, the arithmetic progression given by the recursive equation $t_{n+1}=t_n+\sqrt{2}$`t``n`+1=`t``n`+√2 with $t_1=1$`t`1=1 can be constructed term by term as:

$t_2=t_1+\sqrt{2}=1+\sqrt{2}$`t`2=`t`1+√2=1+√2

$t_3=t_2+\sqrt{2}=\left(1+\sqrt{2}\right)+\sqrt{2}=1+2\sqrt{2}$`t`3=`t`2+√2=(1+√2)+√2=1+2√2

$t_4=t_3+\sqrt{2}=\left(1+2\sqrt{2}\right)+\sqrt{2}=1+3\sqrt{2}$`t`4=`t`3+√2=(1+2√2)+√2=1+3√2

From the first four terms, it is clear that the sequence becomes:

$1,1+\sqrt{2},1+2\sqrt{2},1+3\sqrt{2},...,1+\left(n-1\right)\sqrt{2}+...$1,1+√2,1+2√2,1+3√2,...,1+(`n`−1)√2+...

An interesting sequence is given recursively as $t_{n+1}=t_n+n$`t``n`+1=`t``n`+`n`, with $t_1=0$`t`1=0. To test whether it is an arithmetic sequence, we will consider the first four terms. Here:

$t_1=0$`t`1=0,

$t_2=t_1+2=2$`t`2=`t`1+2=2,

$t_3=t_2+3=2+3=5$`t`3=`t`2+3=2+3=5,

$t_4=t_3+4=5+4=9$`t`4=`t`3+4=5+4=9,

The differences in successive terms is not constant, but rises by $1$1 each time, so there is definitely a pattern to the sequence, but it is not arithmetic.

Consider the first-order recurrence relationship defined by $T_n=T_{n-1}+2$`T``n`=`T``n`−1+2, $T_1=5$`T`1=5.

Determine the next four terms of the sequence, from $T_2$

`T`2 to $T_5$`T`5.Write all four terms on the same line, separated by commas.

Plot the first five terms on the graph below.

Loading Graph...Is the sequence generated from this definition arithmetic or geometric?

Arithmetic

AGeometric

BNeither

C

The first term of an arithmetic sequence is $2$2. The fifth term is $26$26.

Solve for $d$

`d`, the common difference of the sequence.Write a recursive rule for $T_n$

`T``n` in terms of $T_{n-1}$`T``n`−1 which defines this sequence and an initial condition for $T_1$`T`1.Write both parts on the same line separated by a comma.

Zuber is a taxi service that charges a $\$1.50$$1.50 pick-up fee and $\$1.95$$1.95 per kilometre of travel.

What is the total charge for a $10$10 km journey?

We want to describe this situation as a recursive sequence.

To start with, state the initial condition $T_0$

`T`0.Write a recurrence relation for $T_n$

`T``n` in terms of $T_{n-1}$`T``n`−1 which defines the price of a $n$`n`km trip.