We have learned about arithmetic sequences in detail in the previous lesson. It is important to practice these types of questions both with and without the use of a calculator.
We can use a CAS calculator to:
- list the terms of a sequence from the recursive rule or the explicit form. This can help if we need to find later terms in the sequence as listing or calculating can be very time consuming.
- graph the terms of a sequence from the recursive rule and from the explicit form. This can help to see long term patterns and trends.
It is worth revisiting the calculator instructions from the first lesson in this chapter, as well as the examples below to review how to find utilise your calculator to perform these tasks.
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=tn−1+d, where $t_1=a$t1=a or alternatively $t_{n+1}=t_n+d$tn+1=tn+d, where $t_1=a$t1=a
This can be referred to as the recursive rule/equation for the arithmetic progression.
- 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+(n−1)d
This can be referred to as the explicit rule, the general rule or the rule for the $n$nth term.
Select the brand of calculator you use below to work through an example of using a calculator for arithmetic sequences and then try the practice questions.
Casio Classpad
How to use the CASIO Classpad to complete the following tasks regarding arithmetic sequences.
Generate the first $10$10 terms of the sequence with the recursive relationship: $t_n=t_{n-1}+4,t_1=3$tn=tn−1+4,t1=3
Generate the first $10$10 terms of the sequence with the explicit relationship: $t_n=7+3n$tn=7+3n
Find the $100$100th term of the sequence $t_n=t_{n-1}+2.5$tn=tn−1+2.5, $t_1=8$t1=8.
Given the third term of an arithmetic sequence is $20$20 and the $12$12th term is $56$56. Use your calculator to find the explicit rule for the sequence in the form $t_n=a+(n-1)d$tn=a+(n−1)d.
TI Nspire
How to use the TI Nspire to complete the following tasks regarding arithmetic sequences.
Generate the first $10$10 terms of the sequence with the recursive relationship: $t_n=t_{n-1}+4,t_1=3$tn=tn−1+4,t1=3
Generate the first $10$10 terms of the sequence with the explicit relationship: $t_n=7+3n$tn=7+3n
Find the $100$100th term of the sequence $t_n=t_{n-1}+2.5$tn=tn−1+2.5, $t_1=8$t1=8.
Given the third term of an arithmetic sequence is $20$20 and the $12$12th term is $56$56. Use your calculator to find the explicit rule for the sequence in the form $t_n=a+(n-1)d$tn=a+(n−1)d.
Practice questions
Question 1
Consider the sequence defined by $a_1=6$a1=6 and $a_n=a_{n-1}+5$an=an−1+5 for $n\ge2$n≥2.
What is the $21$21st term of the sequence?
What is the $22$22nd term of the sequence?
What is the $23$23rd term of the sequence?
What is the $24$24th term of the sequence?
What is the $25$25th term of the sequence?
Question 2
In an arithmetic progression, $T_5=15$T5=15 and $T_{20}=45$T20=45.
By substituting $T_5=15$T5=15 into the equation $T_n=a+\left(n-1\right)d$Tn=a+(n−1)d, form an equation for $a$a in terms of $d$d.
By substituting $T_{20}=45$T20=45 into the equation $T_n=a+\left(n-1\right)d$Tn=a+(n−1)d, form another equation for $a$a in terms of $d$d.
Hence solve for the value of $d$d.
Hence solve for the value of $a$a.
Find $T_{10}$T10, the $10$10th term in the sequence.
What is the sum of the first $11$11 terms?
Question 3
Consider the following sequence given by the recursive rule.
$T_{n+1}=T_n+4$Tn+1=Tn+4, $T_1=-1$T1=−1
Plot the first six points of the sequence.
Loading Graph...State the explicit rule for $T_n$Tn in terms of $n$n.
State the first position $n$n where the sequence becomes greater than or equal to $500$500.
Question 4
$x+4$x+4, $6x+5$6x+5, and $10x-1$10x−1 are three successive terms in an arithmetic progression.
Determine the value of $x$x.