Writing Expressions for the nth Term of an Arithmetic Sequence
Definitions
- Arithmetic Sequence: A sequence of numbers where the same value is added or subtracted each time.
- Example: 5, 9, 13, 17, 21 (increased by 4 each time)
- Term: Each number in the sequence (e.g., 5 is the first term, 9 is the second term, etc.)
- n: Represents the position of the term in the sequence (e.g., n=1 is the first term)
Expression for the nth Term
- General form:
an + b
- a: The common difference (the value added/subtracted each time)
- b: The term derived from the hypothetical term before the first term
Example
- Sequence: 5, 9, 13, 17, 21
- Common Difference: 4
- Expression:
4n + 1
Calculation
- To find the nth term, plug the value of n into the expression.
- Example for the 3rd term (n=3):
4 * 3 + 1 = 13
- Example for the 50th term (n=50):
4 * 50 + 1 = 201
Determining the Expression for a Sequence
- Find the Common Difference
- Identify the number being added/subtracted each time.
- Calculate the Hypothetical Previous Term (b)
- Subtract the common difference from the first actual term.
- Form the Expression
- Combine the results to form the expression
an + b.
Additional Examples
-
Decreasing Sequence
- Sequence: Decreasing by 5 each time (e.g., 26)
- Common Difference: -5
- Hypothetical Previous Term: 31
- Expression:
-5n + 31 or 31 - 5n
-
Increasing by 1.5
- Sequence: Increasing by 1.5 each time (e.g., starting at 1)
- Common Difference: 1.5
- Hypothetical Previous Term: -0.5
- Expression:
1.5n - 0.5
Verification
- Test the expression with known term positions to ensure correctness.
- Example: To verify
1.5n - 0.5 for the 5th term:
- Calculation:
1.5 * 5 - 0.5 = 7
- Result matches the known term, confirming the expression.*
Conclusion
- Understanding how to derive and use expressions for arithmetic sequences allows quick calculation of any term position.
- Always verify expressions with known terms to ensure accuracy.
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