Math Problem Statement
find the formula in this recurrence problem simple
Solution
The recurrence relation in the image is as follows:
The task is to find the explicit formula for .
Step 1: Write out a few terms of the sequence
We already see the first few terms in the image:
Step 2: Analyze the recurrence relation
The recurrence can be rewritten as:
Let:
This transforms the recurrence into:
Thus, is a geometric sequence with:
Step 3: Substitute back for
Since , we have:
Final Explicit Formula:
Verification:
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Do you want to explore this recurrence further or have any questions? Here are some related questions:
- Can you explain how this sequence grows exponentially and why the term dominates?
- How would the explicit formula change if the multiplier or additive constant in the recurrence were different?
- Can we use this formula to compute terms for very large efficiently?
- What happens if the recurrence is modified to (arbitrary )?
- How can we derive such explicit formulas for other types of recurrence relations?
Tip: Transforming a recurrence into a simpler form (like a geometric sequence) often makes it much easier to find an explicit solution.
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Math Problem Analysis
Mathematical Concepts
Recurrence Relations
Geometric Sequences
Algebra
Formulas
T(n) = -3 * 3^n + 4
Theorems
Geometric Sequence Formula
Suitable Grade Level
Grades 10-12
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