Geometric Sequences Notes and Worksheets - Lindsay Bowden - Free Printable

Here are the step-by-step solutions for each problem on your worksheet.

1. Is the sequence geometric or not? $\{-5, 10, -20...\}$

To check if a sequence is geometric, we look for a common ratio ($r$). This means we divide each term by the one before it.
* $10 \div -5 = -2$
* $-20 \div 10 = -2$

Since the ratio is the same every time, yes, it is geometric. The common ratio is $-2$.

2. Find the next 3 terms in the sequence. $\{100, 50, 25...\}$

First, find the pattern (common ratio):
* $50 \div 100 = 0.5$ (or $\frac{1}{2}$)
* $25 \div 50 = 0.5$ (or $\frac{1}{2}$)

The rule is to multiply by $0.5$ (or divide by 2). Let's find the next three:
1. $25 \times 0.5 = 12.5$
2. $12.5 \times 0.5 = 6.25$
3. $6.25 \times 0.5 = 3.125$

Answer: $12.5, 6.25, 3.125$

5. Write a recursive rule for the $n$th term of the sequence: $\{12, 3, 0.75...\}$

A recursive rule tells you how to get the next number using the previous one.
* First term ($a_1$): $12$
* Common ratio ($r$): $3 \div 12 = 0.25$ (or $\frac{1}{4}$)

Rule:
$a_1 = 12$
$a_n = 0.25 \cdot a_{n-1}$

6. Write an explicit rule for the $n$th term of the sequence: $\{2, 8, 32...\}$. Then find $a_{21}$.

An explicit rule lets you plug in $n$ to find any term directly. The formula is $a_n = a_1 \cdot r^{(n-1)}$.
* First term ($a_1$): $2$
* Common ratio ($r$): $8 \div 2 = 4$

Explicit Rule: $a_n = 2 \cdot 4^{(n-1)}$

Find $a_{21}$:
Plug in $21$ for $n$:
$a_{21} = 2 \cdot 4^{(21-1)}$
$a_{21} = 2 \cdot 4^{20}$
*(Note: $4^{20}$ is a very large number, approximately $1.1 \times 10^{12}$. Unless your teacher wants the exact long number, leaving it as $2 \cdot 4^{20}$ is usually correct. If calculated, it is $2,199,023,255,552$.)*

7. Write an explicit rule for the $n$th term of the sequence: $\{6, 18, 54...\}$. Then find $a_{35}$.

* First term ($a_1$): $6$
* Common ratio ($r$): $18 \div 6 = 3$

Explicit Rule: $a_n = 6 \cdot 3^{(n-1)}$

Find $a_{35}$:
Plug in $35$ for $n$:
$a_{35} = 6 \cdot 3^{(35-1)}$
$a_{35} = 6 \cdot 3^{34}$

8. Write a recursive rule for the $n$th term of the sequence: $\{225, 45, 9...\}$. Then find $a_7$.

* First term ($a_1$): $225$
* Common ratio ($r$): $45 \div 225 = 0.2$ (or $\frac{1}{5}$)

Recursive Rule:
$a_1 = 225$
$a_n = 0.2 \cdot a_{n-1}$

Find $a_7$:
We can just multiply by $0.2$ six times starting from 225:
1. $a_1 = 225$
2. $a_2 = 45$
3. $a_3 = 9$
4. $a_4 = 1.8$
5. $a_5 = 0.36$
6. $a_6 = 0.072$
7. $a_7 = 0.0144$

Answer: $0.0144$

9. Use the sequence to complete the table. $\{2, 4, 8, 16, 32...\}$

The pattern here is multiplying by 2 each time ($r=2$).

Table Completion:
* $n=6$: $32 \times 2 = \mathbf{64}$
* $n=7$: $64 \times 2 = \mathbf{128}$
* $n=8$: $128 \times 2 = \mathbf{256}$

Graphing:
Plot these points on your grid: $(1, 2), (2, 4), (3, 8), (4, 16), (5, 32), (6, 64)...$
*Note: The numbers go up very fast! You will likely need to change the scale on your y-axis (count by 10s or 20s instead of 1s) to fit the larger numbers.*

What do you notice about the graph?
The graph creates a curve that goes up very steeply. It is not a straight line; it shows exponential growth.

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Final Answer:

1. Yes, it is geometric.
2. $12.5, 6.25, 3.125$
5. $a_1 = 12, \quad a_n = 0.25 \cdot a_{n-1}$
6. Rule: $a_n = 2 \cdot 4^{(n-1)}$; $\quad a_{21} = 2 \cdot 4^{20}$
7. Rule: $a_n = 6 \cdot 3^{(n-1)}$; $\quad a_{35} = 6 \cdot 3^{34}$
8. Rule: $a_1 = 225, \quad a_n = 0.2 \cdot a_{n-1}$; $\quad a_7 = 0.0144$
9. Table values for $n=6, 7, 8$ are 64, 128, 256. The graph is a steep upward curve (exponential).