We have learned about arithmetic sequences in detail in our 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.

Examples

Example 1

Consider the sequence defined by a_1 = 6 and a_n = a_{n - 1} + 5 for n \geq 2.

What is the 21st term of the sequence?

Worked Solution

Create a strategy

Using the sequence facility on your calculator, enter the general rule for the sequence along with the initial term and value of n that you want to find.

Apply the idea

By entering the rule, initial term and n=21 you should get:a_{21}=106

Example 2

In an arithmetic progression, T_{5} = 15 and T_{20} = 45.

By substituting T_5=15 into the equation T_n=a+\left(n-1\right)d, form an equation for a in terms of d.

Worked Solution

Create a strategy

Substitute the values of n and T_n into the formula for finding the nth term: T_n=a+\left(n-1\right)d.

Apply the idea

We are given n=5 and T_{5}=15.

\displaystyle T_{n}	\displaystyle =	\displaystyle a+(n-1)\times d	Write the formula
\displaystyle 15	\displaystyle =	\displaystyle a+(5-1)\times d	Substitute the values
\displaystyle 15	\displaystyle =	\displaystyle a+4 d	Evaluate the subtraction
\displaystyle 15-4d	\displaystyle =	\displaystyle a+4d-4d	Subtract 4d from both sides
\displaystyle a	\displaystyle =	\displaystyle 15-4d	Simplify

By substituting T_{20}=45 into the equation T_n=a+\left(n-1\right)d, form an equation for a in terms of d.

Worked Solution

Create a strategy

Substitute the values of n and T_n into the formula for finding the nth term: T_n=a+\left(n-1\right)d.

Apply the idea

We are given n=20 and T_{20}=45.

\displaystyle T_{n}	\displaystyle =	\displaystyle a+(n-1)\times d	Write the formula
\displaystyle 45	\displaystyle =	\displaystyle a+(20-1)\times d	Substitute the values
\displaystyle 45	\displaystyle =	\displaystyle a+19 d	Evaluate the subtraction
\displaystyle 45-19d	\displaystyle =	\displaystyle a+19d-19d	Subtract 19d from both sides
\displaystyle a	\displaystyle =	\displaystyle 45-19d	Simplify

Hence solve for the value of d.

Worked Solution

Create a strategy

Equate the two expressions found from parts (a) and (b).

Apply the idea

From parts (a) and (b) we have found two expression that a is equal to: a=15-4d and a=45-19d. So these expressions must be equal, and we get the equation:

\displaystyle 15-4d

\displaystyle =

\displaystyle 45-19d

Equate the two expressions

By using the equation solver function on your calculator to solve the above equation, you should get:

\displaystyle d

\displaystyle =

\displaystyle 2

Hence solve for the value of a.

Worked Solution

Create a strategy

Substitute the common difference found from part (c) into one of the expressions for a.

Apply the idea

Substitute d=2 into a=15-4d.

\displaystyle a	\displaystyle =	\displaystyle 15-4\times 2	Substitute d=2
	\displaystyle =	\displaystyle 7	Evaluate

Find T_{10}, the 10th term in the sequence.

Worked Solution

Create a strategy

Substitute a, d and n=10 into the formula: T_n=a+\left(n-1\right)d.

Apply the idea

We know that a=7 and d=2.

\displaystyle T_{n}	\displaystyle =	\displaystyle a+(n-1)\times d	Write the formula
	\displaystyle =	\displaystyle 7+(n-1)\times 2	Substitute a and d
\displaystyle T_{10}	\displaystyle =	\displaystyle 7+(10-1)\times2	Substitute n=10
	\displaystyle =	\displaystyle 7+9\times2	Evaluate the subtraction
	\displaystyle =	\displaystyle 25	Evaluate

What is the sum of the first 11 terms?

Worked Solution

Create a strategy

Using your calculator, enter in the general rule for the sequence, T_n=7+(n-1)2, along with the initial term a=7 and then sum the first 11 terms together.

Apply the idea

\text{Sum of terms}=187

Example 3

Consider the following sequence given by the recursive rule.T_{n+1}=T_n+4,\,T_1=-1

Plot the first six points of the sequence.

Worked Solution

Create a strategy

Use the sequence facility in your calculator to plot the first 6 terms of the sequence.

Apply the idea

-2

T_n

State the explicit rule for T_n in terms of n.

Worked Solution

Create a strategy

Find a and d then substitute them into the formula: T_n=a+\left(n-1\right)d.

Apply the idea

We are given T_1=a=-1 and we can see that d=4 because each point on the graph goes up by 4.

\displaystyle T_n	\displaystyle =	\displaystyle a+(n-1)\times d	Write the formula
	\displaystyle =	\displaystyle -1+(n-1)\times 4	Substitute a=-1, d=4
	\displaystyle =	\displaystyle -1+4n-4	Expand the brackets
	\displaystyle =	\displaystyle 4n-5	Combine like terms

State the first position n where the sequence becomes greater than or equal to 500.

Worked Solution

Create a strategy

Use your calculator to generate the sequence in a list and look for the first term that is greater than or equal to 500.

Apply the idea

By doing the above you should find that the 127th term is greater or equal to 500. So:n=127

Example 4

x+4,\,6x+5, and 10x-1 are three successive terms in an arithmetic progression. Determine the value of x.

Worked Solution

Create a strategy

Equate the difference between first and second terms with the difference between the second and third terms.

Apply the idea

The difference between successive terms of an arithmetic progression is always the same. So the difference between x+4 and 6x+5 should be equal to the difference between 6x+5 and 10x-1.

\displaystyle 6x+5-\left(x+4\right)

\displaystyle =

\displaystyle 10x-1-\left(6x+5\right)

Equate the differences

Now we can use the equation solver on our calculator to solve for x to get:

\displaystyle x

\displaystyle =

\displaystyle -7

Idea summary

For any arithmetic sequence with starting value a and common difference 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, where t_1=a or alternatively t_{n+1}=t_n+d, where t_1=a.
Explicit form is a way to express any term in relation to the term number: t_n=a+\left(n-1\right)d This can be referred to as the explicit rule, the general rule or the rule for the nth term.

Outcomes

ACMGM067

use recursion to generate an arithmetic sequence

ACMGM068

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

ACMGM069

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

3.03 Arithmetic Sequences with technology

Ideas

Arithmetic sequences with technology

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

ACMGM067

ACMGM068

ACMGM069

What is Mathspace

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