Math Problem Statement
If an = 3" + n3", n = 0, 1, 2, ..., then an satisfies
(A) an = 3an-1+9an-2
(B) an = 3an-1-9an-2
(C) an = 6an-1 +9an-2
(D) an = 6an-1-9an-2
Solution
Given the sequence for , we want to determine which recurrence relation it satisfies.
Let's first expand the given formula for and observe the behavior for some values of :
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Expression for :
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Expression for :
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Expression for :
We want to find a relation involving , , and . Let's express in terms of and .
Step-by-step Calculation
Let's manipulate these expressions:
Multiplying by 3:
Now, :
Now let's combine these results to express :
Thus, the correct recurrence relation is:
Correct Answer:
The correct option is (A) .
Would you like more details on the steps involved in deriving this solution, or do you have any other questions?
Here are some related questions that might interest you:
- What is the method for finding recurrence relations in sequences?
- How can you determine the closed-form solution for a given recurrence relation?
- What is the characteristic equation of a recurrence relation, and how is it used?
- Can you solve non-homogeneous recurrence relations? If so, how?
- How do you find the initial conditions necessary to uniquely determine a sequence defined by a recurrence relation?
Tip: When working with sequences defined by recurrence relations, always check the base cases and ensure the recurrence fits all known terms.
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Math Problem Analysis
Mathematical Concepts
Sequence
Recurrence relation
Formulas
Recurrence relation formula: an = 3an-1 + 9an-2
Theorems
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Suitable Grade Level
High School