We have learned about geometric sequences in detail in our previous lesson . It is important to practice these types of questions both with and without the use of technology.

Examples

Example 1

In a geometric progression, T_4=-192 and T_7=12\,288.

Find the value of r, the common ratio in the sequence.

Worked Solution

Create a strategy

Find how many common ratios need to be multiplied to get from T_4 to T_{7}.

Apply the idea

There are 3 terms between T_4 and T_{7} with r being multiplied each time: \begin{array}{c} &&\times r&&\times r&&\times r \\ &T_4 & \longrightarrow &T_5 & \longrightarrow &T_6 & \longrightarrow &T_{7}\end{array}

So T_7= T_4 \times r \times r \times r =T_4\times r^3. We can solve this equation for r by substituting the values for T_4 and T_7.

\displaystyle T_7	\displaystyle =	\displaystyle T_4 \times r^3	Write the equation
\displaystyle 12\,288	\displaystyle =	\displaystyle -192 \times r^3	Substitute the values
\displaystyle r^3	\displaystyle =	\displaystyle \dfrac{12\,288}{-192}	Divide both sides by -192
\displaystyle r^3	\displaystyle =	\displaystyle -64	Evaluate the division
\displaystyle r	\displaystyle =	\displaystyle -4	Cube root both sides

Find a, the first term in the progression.

Worked Solution

Create a strategy

Substitute the values of T_4, the fourth term in the sequence, and r, the common ratio, into the formula for finding the nth term: T_n=ar^{n-1}.

Apply the idea

We are given T_4=-192 and have found that r=-4.

\displaystyle T_4	\displaystyle =	\displaystyle a\times r^{4-1}	Substitute n=4
\displaystyle -192	\displaystyle =	\displaystyle a\times (-4)^3	Simplify and substitute r=-4,T_4=-192
\displaystyle -192	\displaystyle =	\displaystyle a\times (-64)	Evaluate the power
\displaystyle a	\displaystyle =	\displaystyle \dfrac{-192}{-64}	Divide both sides by -64
	\displaystyle =	\displaystyle 3	Evaluate

Find an expression for T_n, the general nth term.

Worked Solution

Create a strategy

Substitute the values of a, the first term in sequence, and r, the common ratio, into the formula for finding the nth term: T_n=ar^{n-1}.

Apply the idea

We have found that a=3 and r=-4.

\displaystyle T_n	\displaystyle =	\displaystyle a \times r^{n-1}	Write the formula
	\displaystyle =	\displaystyle 3 \times (-4)^{n-1}	Substitute the values

Example 2

Consider the following sequence.

54 ,\,18 ,\,6,\,2,\,\ldots

If the sequence starts from n=1, plot the first four terms on a graph.

Worked Solution

Create a strategy

Write the ordered pairs for n=1,2,3,4 based on the given sequence and then plot these ordered pairs on a plane with n on the horizontal axis and T_n on the vertical axis.

Apply the idea

T_n

When n=1, T_1=54. This can be expressed as the ordered pair (1,54). Similarly the rest of the ordered pairs are: (2,18),\,(3,6), and (4,2).

Plotting the ordered pairs, we get this graph.

The relationship depicted by this graph is:

linear

exponential

neither

Worked Solution

Create a strategy

Examine the graph from part (a) and visualise what type of function would pass through all the points.

Apply the idea

T_n

If we join the four points with a curve, the curve line gets more shallow as n increases. We can see that the graph is exponential.

So, the correct answer is Option B.

Write the recursive rule for T_n in terms of T_{n-1}, including the initial term T_1.

Worked Solution

Create a strategy

State the initial value T_1 and the rule using the formula T_{n}=rT_{n-1}.

Apply the idea

Since the first y-coordinate is 54, T_1=54.

To get the common ratio r, we divide the second y-coordinate, 18 by the first y-coordinate, 54.

\displaystyle r	\displaystyle =	\displaystyle \dfrac{18}{54}	Divide 54 by 18
	\displaystyle =	\displaystyle \dfrac{1}{3}	Evaluate

This means that each point on the graph has a height of \dfrac{1}{3} of the point before it.

So, the recursive rule is given by:T_n=\frac{1}{3}T_{n-1},T_1=54

What is the sum of the first 10 terms? Round your answer to the nearest whole number.

Worked Solution

Create a strategy

Using your calculator, enter the recursive rule from part (c) along with the initial term and then sum the first 10 terms of the generated sequence.

Apply the idea

\text{Sum of terms}=81

Idea summary

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 may 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.

Outcomes

ACMGM071

use recursion to generate a geometric sequence

ACMGM072

display the terms of a geometric sequence in both tabular and graphical form and demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations

ACMGM073

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

3.05 Geometric sequences with technology

Ideas

Geometric sequences with technology

Examples

Example 1

Example 2

Outcomes

ACMGM071

ACMGM072

ACMGM073

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