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 an=3n+n3na_n = 3^n + n \cdot 3^n for n=0,1,2,n = 0, 1, 2, \ldots, we want to determine which recurrence relation it satisfies.

Let's first expand the given formula for ana_n and observe the behavior for some values of nn:

  1. Expression for ana_n: an=3n+n3na_n = 3^n + n \cdot 3^n

  2. Expression for an1a_{n-1}: an1=3n1+(n1)3n1a_{n-1} = 3^{n-1} + (n-1) \cdot 3^{n-1}

  3. Expression for an2a_{n-2}: an2=3n2+(n2)3n2a_{n-2} = 3^{n-2} + (n-2) \cdot 3^{n-2}

We want to find a relation involving ana_n, an1a_{n-1}, and an2a_{n-2}. Let's express ana_n in terms of an1a_{n-1} and an2a_{n-2}.

Step-by-step Calculation

Let's manipulate these expressions:

an=3n+n3n=(1+n)3na_n = 3^n + n \cdot 3^n = (1 + n)3^n

Multiplying an1a_{n-1} by 3:

3an1=3(3n1+(n1)3n1)=3n+(n1)3n=(1+n1)3n=n3n3a_{n-1} = 3 \left(3^{n-1} + (n-1) \cdot 3^{n-1}\right) = 3^n + (n-1) \cdot 3^n = (1 + n - 1)3^n = n3^n

Now, 9an29a_{n-2}:

9an2=9(3n2+(n2)3n2)=3n+(n2)3n9a_{n-2} = 9 \left(3^{n-2} + (n-2) \cdot 3^{n-2}\right) = 3^n + (n-2) \cdot 3^n

Now let's combine these results to express ana_n:

an=3an1+9an2a_n = 3a_{n-1} + 9a_{n-2}

Thus, the correct recurrence relation is:

an=3an1+9an2a_n = 3a_{n-1} + 9a_{n-2}

Correct Answer:

The correct option is (A) an=3an1+9an2a_n = 3a_{n-1} + 9a_{n-2}.


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:

  1. What is the method for finding recurrence relations in sequences?
  2. How can you determine the closed-form solution for a given recurrence relation?
  3. What is the characteristic equation of a recurrence relation, and how is it used?
  4. Can you solve non-homogeneous recurrence relations? If so, how?
  5. 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