Arithmetic and geometric sequences worksheet - Free Printable

Problem Breakdown and Solution

#### Part 1: Complete the table below

We are given several sequences defined by their general terms \( a_n \). We need to find the values of \( a_1 \), \( a_2 \), and \( a_{10} \) for each sequence.

The sequences are:
1. \( a_n = 2n - 1 \)
2. \( a_n = \frac{4n - 3}{2} \)
3. \( a_n = n^2 - 3n + 5 \)
4. \( a_n = 2^{n-1} \)
5. \( a_n = (-3)^n \)

##### Sequence 1: \( a_n = 2n - 1 \)

- \( a_1 \): Substitute \( n = 1 \):
\[
a_1 = 2(1) - 1 = 2 - 1 = 1
\]

- \( a_2 \): Substitute \( n = 2 \):
\[
a_2 = 2(2) - 1 = 4 - 1 = 3
\]

- \( a_{10} \): Substitute \( n = 10 \):
\[
a_{10} = 2(10) - 1 = 20 - 1 = 19
\]

##### Sequence 2: \( a_n = \frac{4n - 3}{2} \)

- \( a_1 \): Substitute \( n = 1 \):
\[
a_1 = \frac{4(1) - 3}{2} = \frac{4 - 3}{2} = \frac{1}{2}
\]

- \( a_2 \): Substitute \( n = 2 \):
\[
a_2 = \frac{4(2) - 3}{2} = \frac{8 - 3}{2} = \frac{5}{2}
\]

- \( a_{10} \): Substitute \( n = 10 \):
\[
a_{10} = \frac{4(10) - 3}{2} = \frac{40 - 3}{2} = \frac{37}{2}
\]

##### Sequence 3: \( a_n = n^2 - 3n + 5 \)

- \( a_1 \): Substitute \( n = 1 \):
\[
a_1 = (1)^2 - 3(1) + 5 = 1 - 3 + 5 = 3
\]

- \( a_2 \): Substitute \( n = 2 \):
\[
a_2 = (2)^2 - 3(2) + 5 = 4 - 6 + 5 = 3
\]

- \( a_{10} \): Substitute \( n = 10 \):
\[
a_{10} = (10)^2 - 3(10) + 5 = 100 - 30 + 5 = 75
\]

##### Sequence 4: \( a_n = 2^{n-1} \)

- \( a_1 \): Substitute \( n = 1 \):
\[
a_1 = 2^{1-1} = 2^0 = 1
\]

- \( a_2 \): Substitute \( n = 2 \):
\[
a_2 = 2^{2-1} = 2^1 = 2
\]

- \( a_{10} \): Substitute \( n = 10 \):
\[
a_{10} = 2^{10-1} = 2^9 = 512
\]

##### Sequence 5: \( a_n = (-3)^n \)

- \( a_1 \): Substitute \( n = 1 \):
\[
a_1 = (-3)^1 = -3
\]

- \( a_2 \): Substitute \( n = 2 \):
\[
a_2 = (-3)^2 = 9
\]

- \( a_{10} \): Substitute \( n = 10 \):
\[
a_{10} = (-3)^{10} = 59049
\]

##### Completed Table

| \( a_n \) | \( a_1 \) | \( a_2 \) | \( a_{10} \) |
|-------------------|-----------|-----------|--------------|
| \( a_n = 2n - 1 \) | 1 | 3 | 19 |
| \( a_n = \frac{4n - 3}{2} \) | \(\frac{1}{2}\) | \(\frac{5}{2}\) | \(\frac{37}{2}\) |
| \( a_n = n^2 - 3n + 5 \) | 3 | 3 | 75 |
| \( a_n = 2^{n-1} \) | 1 | 2 | 512 |
| \( a_n = (-3)^n \) | -3 | 9 | 59049 |

---

#### Part 2: Determine if the sequence is arithmetic or geometric. Find the common ratio or the difference.

We are given several sequences, and we need to determine whether each is arithmetic or geometric. For arithmetic sequences, we find the common difference (\( d \)). For geometric sequences, we find the common ratio (\( r \)).

##### Sequence 1: \( -9, -109, -209, -309, \ldots \)

- Check for arithmetic sequence: Calculate the differences between consecutive terms:
\[
-109 - (-9) = -109 + 9 = -100
\]
\[
-209 - (-109) = -209 + 109 = -100
\]
The differences are constant (\( d = -100 \)), so this is an arithmetic sequence.

- Common difference: \( d = -100 \)

##### Sequence 2: \( 1, 0.5, 0, -0.5, \ldots \)

- Check for arithmetic sequence: Calculate the differences between consecutive terms:
\[
0.5 - 1 = -0.5
\]
\[
0 - 0.5 = -0.5
\]
The differences are constant (\( d = -0.5 \)), so this is an arithmetic sequence.

- Common difference: \( d = -0.5 \)

##### Sequence 3: \( -1, 5, -25, -125, \ldots \)

- Check for geometric sequence: Calculate the ratios between consecutive terms:
\[
\frac{5}{-1} = -5
\]
\[
\frac{-25}{5} = -5
\]
The ratios are constant (\( r = -5 \)), so this is a geometric sequence.

- Common ratio: \( r = -5 \)

##### Sequence 4: \( 4, 12, 36, 108, \ldots \)

- Check for geometric sequence: Calculate the ratios between consecutive terms:
\[
\frac{12}{4} = 3
\]
\[
\frac{36}{12} = 3
\]
The ratios are constant (\( r = 3 \)), so this is a geometric sequence.

- Common ratio: \( r = 3 \)

##### Sequence 5: \( -20, -29, -38, -47, \ldots \)

- Check for arithmetic sequence: Calculate the differences between consecutive terms:
\[
-29 - (-20) = -29 + 20 = -9
\]
\[
-38 - (-29) = -38 + 29 = -9
\]
The differences are constant (\( d = -9 \)), so this is an arithmetic sequence.

- Common difference: \( d = -9 \)

##### Sequence 6: \( 0.5, 1, 2, 4, \ldots \)

- Check for geometric sequence: Calculate the ratios between consecutive terms:
\[
\frac{1}{0.5} = 2
\]
\[
\frac{2}{1} = 2
\]
The ratios are constant (\( r = 2 \)), so this is a geometric sequence.

- Common ratio: \( r = 2 \)

##### Completed Table

| Sequence | Arithmetic or Geometric | Ratio or Difference |
|------------------------|--------------------------|----------------------|
| \( -9, -109, -209, \ldots \) | Arithmetic | \( d = -100 \) |
| \( 1, 0.5, 0, -0.5, \ldots \) | Arithmetic | \( d = -0.5 \) |
| \( -1, 5, -25, -125, \ldots \) | Geometric | \( r = -5 \) |
| \( 4, 12, 36, 108, \ldots \) | Geometric | \( r = 3 \) |
| \( -20, -29, -38, -47, \ldots \) | Arithmetic | \( d = -9 \) |
| \( 0.5, 1, 2, 4, \ldots \) | Geometric | \( r = 2 \) |

---

Final Answer

\[
\boxed{
\begin{array}{c|c|c|c}
a_n & a_1 & a_2 & a_{10} \\
\hline
a_n = 2n - 1 & 1 & 3 & 19 \\
a_n = \frac{4n - 3}{2} & \frac{1}{2} & \frac{5}{2} & \frac{37}{2} \\
a_n = n^2 - 3n + 5 & 3 & 3 & 75 \\
a_n = 2^{n-1} & 1 & 2 & 512 \\
a_n = (-3)^n & -3 & 9 & 59049 \\
\end{array}
}
\]
\[
\boxed{
\begin{array}{c|c|c}
\text{Sequence} & \text{Arithmetic or Geometric} & \text{Ratio or Difference} \\
\hline
-9, -109, -209, \ldots & \text{Arithmetic} & d = -100 \\
1, 0.5, 0, -0.5, \ldots & \text{Arithmetic} & d = -0.5 \\
-1, 5, -25, -125, \ldots & \text{Geometric} & r = -5 \\
4, 12, 36, 108, \ldots & \text{Geometric} & r = 3 \\
-20, -29, -38, -47, \ldots & \text{Arithmetic} & d = -9 \\
0.5, 1, 2, 4, \ldots & \text{Geometric} & r = 2 \\
\end{array}
}
\]