An arithmetic sequence with fractions involves finding a pattern where a constant value (the common difference) is added to each term to get the next term, and at least one of the terms or the common difference is a fraction. Here's how to work with them:
1. Identify the First Term (a₁) and the Common Difference (d)
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The first term (a₁) is simply the first number in the sequence. This might be a fraction or a whole number.
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The common difference (d) is the constant value added to each term. To find it, subtract any term from the term that follows it. If the terms are fractions, make sure to perform the subtraction correctly (find a common denominator if necessary).
For example, given a sequence like 1/2, 1, 3/2, 2,...:
- a₁ = 1/2
- d = 1 - 1/2 = 1/2 (or 3/2 - 1 = 1/2, or 2 - 3/2 = 1/2)
2. Finding a Specific Term (aₙ)
You can find any term in the sequence using the formula:
aₙ = a₁ + (n - 1)d
Where:
- aₙ = the nth term you want to find
- a₁ = the first term
- n = the term number you want to find (e.g., 5th term, 10th term)
- d = the common difference
Example: Find the 10th term in the sequence 1/2, 1, 3/2, 2,...
- a₁ = 1/2
- d = 1/2
- n = 10
a₁₀ = (1/2) + (10 - 1)(1/2)
a₁₀ = (1/2) + (9)(1/2)
a₁₀ = (1/2) + (9/2)
a₁₀ = 10/2 = 5
Therefore, the 10th term is 5.
3. Working with Fractions: Key Considerations
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Common Denominators: When adding or subtracting fractions to find the common difference or calculate a specific term, ensure the fractions have a common denominator.
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Simplifying: Simplify your answers whenever possible.
4. Example Sequence breakdown
The sequence 1/2, 1, 3/2... shows the common difference is 1/2:
- 1/2 + 1/2 = 1
- 1 + 1/2 = 3/2
- 3/2 + 1/2 = 2
Summary
Arithmetic sequences with fractions follow the same principles as those with whole numbers. The critical steps involve identifying the first term, calculating the common difference (paying close attention to fraction arithmetic), and then applying the formula aₙ = a₁ + (n - 1)d to find the value of a specific term.
