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3.03 Arithmetic progressions - calculator free

Lesson

 

A sequence in which each term changes from the last by adding a constant amount is called an arithmetic sequence. We refer to the constant the terms are changing by as the common difference, which will result from subtracting any two successive terms $\left(t_{n+1}-t_n\right)$(tn+1tn).

The progression $-3,5,13,21,\ldots$3,5,13,21, is an arithmetic progression with a common difference of $8$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.

We denote the first term by the letter $a$a and the common difference by the letter $d$d. Since, $t_2=t_1+d$t2=t1+d$t_3=t_2+d$t3=t2+d and so on, we can write any arithmetic sequence as the recurrence relation:

$t_n=t_{n-1}+d,t_1=a$tn=tn1+d,t1=a

If we consider the recurrence relation $t_n=t_{n-1}+2,t_1=5$tn=tn1+2,t1=5, this recurrence relation starts with $5$5 and we add $2$2 to find the next term, therefore the sequence is $5,7,9,11,13,15...$5,7,9,11,13,15...

We can also find an explicit formula in terms of $a$a and $d$d, this is useful for finding the $n$nth term without listing the sequence.

Let's look at a table of a concrete example to see the pattern for the explicit formula. For the sequence $-3,5,13,21,\ldots$3,5,13,21,, we have starting term of $-3$3 and a common difference of $8$8, that is $a=-3$a=3 and $d=8$d=8. A table of the sequence is show below:

$n$n $t_n$tn Pattern
$1$1 $-3$3 $-3$3
$2$2 $5$5 $-3+8$3+8
$3$3 $13$13 $-3+2\times8$3+2×8
$4$4 $21$21 $-3+3\times8$3+3×8
...    
$n$n $t_n$tn $-3+(n-1)\times8$3+(n1)×8

 

 

The pattern starts to become clear and we could guess that the tenth term becomes $t_{10}=69=-3+9\times8$t10=69=3+9×8  and the one-hundredth term $t_{100}=789=-3+99\times8$t100=789=3+99×8. And following the pattern, the explicit formula for the $n$nth term is $t_n=-3+(n-1)\times8$tn=3+(n1)×8.

We could create a similar table for the arithmetic progression with starting value $a$a and common difference $d$d and we would observe the same pattern. Hence,  generating the explicit rule for any arithmetic sequence is given by:

 $t_n=a+\left(n-1\right)d$tn=a+(n1)d

 

Forms of arithmetic sequences

For any arithmetic sequence with starting value $a$a and common difference $d$d, we can express it in either of the following two forms:

  • Recursive form is a way to express any term in relation to the previous term:

$t_n=t_{n-1}+d$tn=tn1+d, where $t_1=a$t1=a

  • Explicit form is a way to express any term in relation to the term number

$t_n=a+\left(n-1\right)d$tn=a+(n1)d

 

Worked examples

Example 1

For the sequence  $87,80,73,66...$87,80,73,66..., find and explicit rule for the $n$nth term and hence, find the $30$30th term. 

Think: Check that the sequence is arithmetic, does each term differ from the last by a constant? Then write down the the starting value $a$a and common difference $d$d and substitute these into the general form: $t_n=a+(n-1)d$tn=a+(n1)d

Do: Each term is a decrease from the last by $7$7. So we have an arithmetic sequence with: $a=87$a=87 and $d=-7$d=7. The general formula for this sequence is: $t_n=87+\left(n-1\right)\times\left(-7\right)$tn=87+(n1)×(7) or $t_n=87-7(n-1)$tn=877(n1).

Hence, the $30$30th term is:$t_n=87+\left(30-1\right)\times\left(-7\right)$tn=87+(301)×(7)$t_{30}=87-7\times29=-116$t30=877×29=116.

Example 2

For the sequence $10,14,18,22,26,...$10,14,18,22,26,..., find $n$n if the $n$nth term is $186$186.

Think: Find a general rule for the sequence, substitute in $186$186 for $t_n$tn and rearrange for $n$n. Finding $n$nis determining which term position has a value of 186.

Do: This is an arithmetic sequence with $a=10$a=10 and common difference $d=4$d=4. Hence, the general rule is: $t_n=10+\left(n-1\right)\times4$tn=10+(n1)×4, we can simplify this to $t_n=6+4n$tn=6+4n, by expanding brackets and collecting like terms. Substituting $t_n=186$tn=186, we get:

$186$186 $=$= $6+4n$6+4n
$\therefore4n$4n $=$= $180$180
$n$n $=$= $45$45

Hence, the $45$45th term in the sequence is $186$186.

Example 3

If an arithmetic sequence has $T_5=38$T5=38 and $T_9=66$T9=66, find the recurrence relation for the sequence. 

Think: Consider how many times the common difference would need to be added to get from $T_5$T5 to $T_9$T9.

$T_5$T5   $T_6$T6   $T_7$T7   $T_8$T8   $T_9$T9
         
  $+d$+d   $+d$+d   $+d$+d   $+d$+d  

 

Do: The common difference needs to be added to $T_5$T5 four times to get to $T_9$T9. Therefore $T_5+4d=T_9$T5+4d=T9 so $38+4d=66$38+4d=66, this can be rearranged to solve for $d$d. Alternatively, we can use logic, an increase from $38$38 to $66$66 is made up of $4$4 common differences, so $d=(66-38)/4$d=(6638)/4

The common difference found to be $7$7, then we know that, using equation  $a+4\times7=38$a+4×7=38 and so $a$a is $10$10. The recurrence relation for this sequence is given by:  

$T_n=T_{n-1}+7,T_1=10$Tn=Tn1+7,T1=10

 

Practice questions

Question 1

The $n$nth term in an arithmetic progression is given by the formula $T_n=15+5\left(n-1\right)$Tn=15+5(n1).

  1. Determine $a$a, the first term in the arithmetic progression.

  2. Determine $d$d, the common difference.

  3. Determine $T_9$T9, the $9$9th term in the sequence.

Question 2

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

  1. Solve for $d$d, the common difference of the sequence.

  2. Write a recursive rule for $T_n$Tn in terms of $T_{n-1}$Tn1 which defines this sequence and an initial condition for $T_1$T1.

    Write both parts on the same line separated by a comma.

Question 3

An arithmetic progression has a first term of $T_1=a$T1=a and a common difference of $d$d.

Two of the terms in the sequence are $T_7=43$T7=43 and $T_{14}=85$T14=85.

  1. Determine $d$d, the common difference.

  2. Determine $a$a, the first term in the sequence.

  3. State the equation for $T_n$Tn, the $n$nth term in the sequence.

  4. Hence find $T_{25}$T25, the $25$25th term in the sequence.

 

Arithmetic sequences in tables and graphs

For any arithmetic sequence in the general form given by $t_n=a+\left(n-1\right)d$tn=a+(n1)d, we can expand the bracket and collect like terms, creating a new generating rule of the form $t_n=dn+k$tn=dn+k where $d$d and $k$k are constants. For example, the rule $t_n=5+\left(n-1\right)\times2$tn=5+(n1)×2 is equivalent to $t_n=2n+3$tn=2n+3. This is in the form of the equation of a straight line $\left(y=mx+c\right)$(y=mx+c), so if an arithmetic sequence is plotted as a series of points, they will all lie on a straight line with the gradient being the common difference. This makes sense since we have a constant rate of change.

The first term is represented by the point shown at $n=1$n=1$t_1=5$t1=5 and we can see the gradient here is the common difference $d=2$d=2

We could also be expected to recognise an arithmetic sequence from a table, such as:

$n$n $1$1 $2$2 $3$3 $4$4 $5$5
$t_n$tn $5$5 $7$7 $9$9 $11$11 $13$13

Here, we can read off the initial term $t_1=5$t1=5 and the common difference can be seen in step between the $t_n$tn values in the second row.

This interactive tool can show us how arithmetic sequences are actually linear relationships.

 

Practice questions

Question 4

The $n$nth term of an arithmetic progression is given by the equation $T_n=12+4\left(n-1\right)$Tn=12+4(n1).

  1. Complete the table of values.

    $n$n $1$1 $2$2 $3$3 $4$4 $10$10
    $T_n$Tn $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. By how much are consecutive terms in the sequence increasing?

  3. Plot the points in the table on the graph.

    Loading Graph...

  4. If the points on the graph were joined, they would form:

    a straight line

    A

    a curved line

    B

Question 5

The plotted points represent terms in an arithmetic sequence:

Loading Graph...

  1. Complete the table of values for the given points.

    $n$n $1$1 $2$2 $3$3 $4$4
    $T_n$Tn $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Identify $d$d, the common difference between consecutive terms.

  3. Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.

  4. Find the $15$15th term of the sequence.

Question 6

The given table of values represents terms in an arithmetic sequence.

$n$n $1$1 $2$2 $3$3 $4$4
$T_n$Tn $9$9 $17$17 $25$25 $33$33
  1. Identify $d$d, the common difference between consecutive terms.

  2. Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.

  3. Find the $15$15th term of the sequence.

Outcomes

3.2.1

use recursion to generate an arithmetic sequence

3.2.2

display the terms of an arithmetic sequence in both tabular and graphical form and demonstrate that arithmetic sequences can be used to model linear growth and decay in discrete situations

3.2.3

deduce a rule for the nth term of a particular arithmetic sequence from the pattern of the terms in an arithmetic sequence, and use this rule to make predictions

3.2.4

use arithmetic sequences to model and analyse practical situations involving linear growth or decay

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