Recursive rules
A CAS calculator can be used to generate the terms of a sequence given a recursive rule. When problem solving involving sequences we can make effective use of our calculator to:
- List the terms of the sequence
- Find a particular term in a sequence
- Calculate the sum of a given set of terms from a sequence
- Graph the sequence to observe patterns in the behaviour of the sequence
Select the brand of calculator you use below to work through an example of using a calculator with the recursive sequence
$t_n=t_{n-1}+5$tn=tn−1+5, $t_1=2$t1=2
Note: this rule can be described in words as "next term is previous term plus $5$5; with initial term equal to $2$2".
As $t_{n+1}$tn+1 is the next term after $t_n$tn, an alternative way to write the rule is $t_{n+1}=t_n+5$tn+1=tn+5, $t_1=2$t1=2
Casio Classpad
How to use the CASIO Classpad to complete the following tasks regarding recursive rules
Generate the first $10$10 terms of the sequence with the recursive relationship: $t_n=t_{n-1}+5,t_1=2$tn=tn−1+5,t1=2
Find the $100$100th term of the given sequence.
Find the sum of the first $10$10 terms.
Plot the sequence and describe the long-term behaviour.
TI Nspire
How to use the TI Nspire to complete the following tasks regarding recursive rules
Generate the first $10$10 terms of the sequence with the recursive relationship: $t_n=t_{n-1}+5,t_1=2$tn=tn−1+5,t1=2
Find the $100$100th term of the given sequence.
Find the sum of the first $10$10 terms.
Plot the sequence and describe the long-term behaviour.
Practice questions
Question 1
If $T_n$Tn describes the $n$nth term in the following sequence, what is $T_3+T_5$T3+T5?
$6,-8,9,-10,11,\ldots$6,−8,9,−10,11,…
Question 2
What is the $53$53rd term in the following sequence?
$2,3.5,5,6.5,8,9.5,\ldots$2,3.5,5,6.5,8,9.5,…
Question 3
Consider the sequence which has a first term of $11$11 and a second term of $22$22, and subsequent terms are found by adding the two previous terms.
What is the recursive rule for the sequence?
$T_n=2\left(n-1\right)$Tn=2(n−1)
A$T_n=2T_{n-1}$Tn=2Tn−1
B$T_n=T_{n-1}T_{n-2}$Tn=Tn−1Tn−2
C$T_n=T_{n-1}+T_{n-2}$Tn=Tn−1+Tn−2
DState the first five terms of the sequence.
Enter each term on the same line, separated by commas.
Question 4
Consider the following sequences.
$T_{n+1}=2.5T_n$Tn+1=2.5Tn, $T_1=5$T1=5 and $A_{n+1}=3.5A_n$An+1=3.5An, $A_1=5$A1=5
Compare the first $10$10 terms of each of these sequence on a graph to determine which sequence is increasing faster.
$T_{n+1}=2.5T_n$Tn+1=2.5Tn, $T_1=5$T1=5
A$A_{n+1}=3.5A_n$An+1=3.5An, $A_1=5$A1=5
B
Question 5
Consider the following sequence.
$T_n=T_{n-1}+5$Tn=Tn−1+5, $T_1=10$T1=10
Find $T_{40}$T40.
Find the first term greater than or equal to $300$300.
If $S_n$Sn is the sum of the first $n$n terms, find $S_{30}$S30.
Starting from $n=1$n=1, find the minimum number of terms for the sum to be greater than $5000$5000.
Explicit rules
A CAS calculator can also be used to generate the terms of a sequence given an explicit rule. Consider the following rule for the $n$nth term of the sequence, $T_n=n^2+1$Tn=n2+1 and select your brand of calculator below to view the problem and steps to follow in your calculator.
Casio Classpad
How to use the CASIO Classpad to complete the following tasks regarding explicit sequences
Generate the first $10$10 terms of the sequence with the explicit rule: $t_n=n^2+1$tn=n2+1.
Find the $100$100th term of the given sequence.
Find the sum of the first $10$10 terms.
Plot the sequence and describe the long-term behaviour.
Bonus tip.
TI Nspire
How to use the TI Nspire to complete the following tasks regarding explicit sequences
Generate the first $10$10 terms of the sequence with the explicit rule: $t_n=n^2+1$tn=n2+1.
Find the $100$100th term of the given sequence.
Find the sum of the first $10$10 terms.
Plot the sequence and describe the long-term behaviour.
Practice questions
Question 6
For the following explicit rule which starts at $n=1$n=1, what is the sum of the terms from the $20$20th to the $30$30th term inclusive?
$T_n=\left(0.2n\right)^2+5n$Tn=(0.2n)2+5n
Question 7
Consider the following sequence starting at $n=1$n=1.
$T_n=5-3n$Tn=5−3n
Find $T_{30}$T30.
Find the first term less than $-150$−150.
If $S_n$Sn is the sum of the first $n$n terms, find $S_{15}$S15.
Starting from $n=1$n=1, find the minimum number of terms required for the sum to be less than $-600$−600.