An Introduction to Geometric Sequences (VCE SSCE General Mathematics): Revision Notes

Worked Example: An Increasing Sequence

Consider the sequence: $2, 6, 18, 54, \ldots$

In this sequence:

• We start with $2$
• To get the next term, we multiply by $3$: $2 \times 3 = 6$
• Then multiply again by $3$: $6 \times 3 = 18$
• And again: $18 \times 3 = 54$

The common ratio is $R = 3$.

Worked Example: A Decreasing Sequence

Consider the sequence: $120, 60, 30, 15, \ldots$

In this sequence:

• We start with $120$
• Each new term is made by halving: $120 \times \frac{1}{2} = 60$
• Then: $60 \times \frac{1}{2} = 30$
• And: $30 \times \frac{1}{2} = 15$

The common ratio is $R = \frac{1}{2}$. Note that dividing by $2$ is the same as multiplying by $\frac{1}{2}$.

Worked Example: Finding $R$ in an Increasing Sequence

Question: Find the common ratio in the sequence $3, 12, 48, 192, \ldots$

Solution:

Step 1: Calculate the common ratio using the first two terms

$R = \frac{t_1}{t_0} = \frac{12}{3} = 4$

Step 2: Verify by checking if multiplying by $4$ produces each term

• $3 \times 4 = 12$ ✓
• $12 \times 4 = 48$ ✓
• $48 \times 4 = 192$ ✓

Step 3: State the answer

The common ratio is $4$.

Worked Example: Finding $R$ in a Decreasing Sequence

Question: Find the common ratio in the sequence $16, 8, 4, 2, \ldots$

Solution:

Step 1: Calculate the common ratio

$R = \frac{t_1}{t_0} = \frac{8}{16} = \frac{1}{2}$

Step 2: Verify by checking

• $16 \times \frac{1}{2} = 8$ ✓
• $8 \times \frac{1}{2} = 4$ ✓
• $4 \times \frac{1}{2} = 2$ ✓

Step 3: State the answer

The common ratio is $\frac{1}{2}$.

Worked Example: Testing if a Sequence is Geometric (Part a)

Question a: Is $2, 10, 50, 250, \ldots$ a geometric sequence?

Solution:

Step 1: Calculate the ratios between successive terms

$R = \frac{t_1}{t_0} = \frac{10}{2} = 5$

$R = \frac{t_2}{t_1} = \frac{50}{10} = 5$

$R = \frac{t_3}{t_2} = \frac{250}{50} = 5$

Step 2: Compare the ratios

All ratios equal $5$, so the common ratio is $5$.

Step 3: Conclusion

The sequence is geometric because the ratio is constant.

Worked Example: Testing if a Sequence is Geometric (Part b)

Question b: Is $3, 6, 18, 36, \ldots$ a geometric sequence?

Solution:

Step 1: Calculate the ratios

$R = \frac{t_1}{t_0} = \frac{6}{3} = 2$

$R = \frac{t_2}{t_1} = \frac{18}{6} = 3$

$R = \frac{t_3}{t_2} = \frac{36}{18} = 2$

Step 2: Compare the ratios

The ratios are different: $2, 3, 2$.

Step 3: Conclusion

The sequence is not geometric because the ratios are not constant.

Worked Example: Generating Terms with a Calculator

Generate the first six terms of $1, 3, 9, 27, \ldots$

Steps:

• Enter $1$ and press ENTER
• Type $\times 3$ and press ENTER
• Each additional press of ENTER generates the next term: $3, 9, 27, 81, 243$

The first six terms are: $1, 3, 9, 27, 81, 243$.

Worked Example: Graphing an Increasing Sequence

Question: Graph the geometric sequence $2, 6, 18, \ldots$

Solution:

Step 1: Find the next term

$R = \frac{6}{2} = 3$

The next term is $18 \times 3 = 54$.

Step 2: Create a table of values

Step 3: Plot the graph

On the graph:

• The horizontal axis ($n$) represents the position number
• The vertical axis ($t_n$) represents the value of the term
• Plot each point from the table

Step 4: Describe the graph

The values lie along a curve, and they are increasing. This exponential growth pattern is characteristic of geometric sequences with $R > 1$.

Worked Example: Graphing a Decreasing Sequence

Question: Graph the geometric sequence $32, 16, 8, \ldots$

Solution:

Step 1: Find the common ratio and next term

$R = \frac{16}{32} = \frac{1}{2}$

The next term is $8 \times \frac{1}{2} = 4$.

Step 2: Create a table of values

Step 3: Plot the graph

Step 4: Describe the graph

The graph is a curve with values decreasing and approaching zero. This exponential decay pattern is characteristic of geometric sequences with $0 < R < 1$.