First Term and Common Difference of an Arithmetic Sequence Worksheets - Free Printable

Problem: Determine the first term \( a \) and the common difference \( d \) for each arithmetic sequence.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (\( d \)). The general form of an arithmetic sequence is:

\[
a, a + d, a + 2d, a + 3d, \ldots
\]

where:
- \( a \) is the first term,
- \( d \) is the common difference.

To solve each part:
1. Identify the first term \( a \).
2. Calculate the common difference \( d \) by subtracting any term from the next term in the sequence.

Let's solve each sequence step by step.

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1) Sequence: \( -700, -750, -800, -850, -900, \ldots \)

- First term (\( a \)): The first number in the sequence is \( -700 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = -750 - (-700) = -750 + 700 = -50
\]

Thus, \( a = -700 \) and \( d = -50 \).

---

2) Sequence: \( 9.1, 2.6, -3.9, -10.4, -16.9, \ldots \)

- First term (\( a \)): The first number in the sequence is \( 9.1 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = 2.6 - 9.1 = -6.5
\]

Thus, \( a = 9.1 \) and \( d = -6.5 \).

---

3) Sequence: \( 27.4, 23.4, 19.4, 15.4, 11.4, \ldots \)

- First term (\( a \)): The first number in the sequence is \( 27.4 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = 23.4 - 27.4 = -4
\]

Thus, \( a = 27.4 \) and \( d = -4 \).

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4) Sequence: \( 1, \frac{3}{2}, \frac{1}{2}, 0, \ldots \)

- First term (\( a \)): The first number in the sequence is \( 1 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = \frac{3}{2} - 1 = \frac{3}{2} - \frac{2}{2} = \frac{1}{2}
\]

Thus, \( a = 1 \) and \( d = \frac{1}{2} \).

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5) Sequence: \( \frac{2}{5}, \frac{16}{15}, \frac{26}{15}, \frac{12}{5}, \frac{46}{15}, \ldots \)

- First term (\( a \)): The first number in the sequence is \( \frac{2}{5} \).
- Common difference (\( d \)): Subtract the first term from the second term. First, express all terms with a common denominator (15):
\[
\frac{2}{5} = \frac{6}{15}, \quad \frac{16}{15} = \frac{16}{15}, \quad \frac{26}{15} = \frac{26}{15}, \quad \frac{12}{5} = \frac{36}{15}, \quad \frac{46}{15} = \frac{46}{15}
\]
Now, calculate \( d \):
\[
d = \frac{16}{15} - \frac{6}{15} = \frac{10}{15} = \frac{2}{3}
\]

Thus, \( a = \frac{2}{5} \) and \( d = \frac{2}{3} \).

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6) Sequence: \( 2, 9, 16, 23, 30, \ldots \)

- First term (\( a \)): The first number in the sequence is \( 2 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = 9 - 2 = 7
\]

Thus, \( a = 2 \) and \( d = 7 \).

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7) Sequence: \( 18, 9, 0, -9, -18, \ldots \)

- First term (\( a \)): The first number in the sequence is \( 18 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = 9 - 18 = -9
\]

Thus, \( a = 18 \) and \( d = -9 \).

---

8) Sequence: \( \frac{1}{3}, \frac{10}{3}, \frac{19}{3}, \frac{28}{3}, \frac{37}{3}, \ldots \)

- First term (\( a \)): The first number in the sequence is \( \frac{1}{3} \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = \frac{10}{3} - \frac{1}{3} = \frac{9}{3} = 3
\]

Thus, \( a = \frac{1}{3} \) and \( d = 3 \).

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9) Sequence: \( \sqrt{2}, -\sqrt{6}, -\sqrt{18}, -\sqrt{242}, -\sqrt{450}, \ldots \)

- First term (\( a \)): The first number in the sequence is \( \sqrt{2} \).
- Common difference (\( d \)): Simplify the terms to identify the pattern:
\[
\sqrt{2}, -\sqrt{6}, -\sqrt{18}, -\sqrt{242}, -\sqrt{450}
\]
Notice that:
\[
\sqrt{6} = \sqrt{2 \cdot 3}, \quad \sqrt{18} = \sqrt{2 \cdot 9}, \quad \sqrt{242} = \sqrt{2 \cdot 121}, \quad \sqrt{450} = \sqrt{2 \cdot 225}
\]
The sequence can be rewritten as:
\[
\sqrt{2}, -\sqrt{2 \cdot 3}, -\sqrt{2 \cdot 9}, -\sqrt{2 \cdot 121}, -\sqrt{2 \cdot 225}
\]
The terms are not arithmetic in their current form, so we need to re-evaluate. However, if we assume the sequence is intended to be arithmetic in its simplified form, we can approximate or check for a pattern. For now, let's assume the sequence is not strictly arithmetic based on the given terms.

Thus, \( a = \sqrt{2} \) and \( d \) cannot be determined as the sequence is not arithmetic.

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10) Sequence: \( -33, -48, -63, -78, -93, \ldots \)

- First term (\( a \)): The first number in the sequence is \( -33 \).
- Common difference (\( d \)): Subtract the first term from the second term:
\[
d = -48 - (-33) = -48 + 33 = -15
\]

Thus, \( a = -33 \) and \( d = -15 \).

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & a = -700, \, d = -50 \\
2) & a = 9.1, \, d = -6.5 \\
3) & a = 27.4, \, d = -4 \\
4) & a = 1, \, d = \frac{1}{2} \\
5) & a = \frac{2}{5}, \, d = \frac{2}{3} \\
6) & a = 2, \, d = 7 \\
7) & a = 18, \, d = -9 \\
8) & a = \frac{1}{3}, \, d = 3 \\
9) & a = \sqrt{2}, \, d = \text{not arithmetic} \\
10) & a = -33, \, d = -15 \\
\end{array}
}
\]