How to Write a Formula for a Recursive Sequence

The formula of a recursive sequence is as follows:

\(a_n = f(a_{(n-1)})\), where a_n is the nth term of the sequence and \(a_{(n-1)}\) is the \((n-1)\)th term of the sequence. The function \(f(a_{(n-1)})\) is called the return relation and describes the relationship in the sequence.

Step-by-step to find the general formula for the recursive sequence

To find the general formula for the recursive sequence, follow the step-by-step guide below:

1. Start by identifying the first few terms of the recursive sequence.
2. Attempt to find a pattern in the terms, such as a specific difference or ratio between consecutive terms.
3. Use the pattern identified in step 2 to express the nth term of the sequence in terms of the previous terms.
4. Solve for the explicit formula using the recursive formula found in step 3 and the initial conditions of the sequence.
5. Test the explicit formula using the first few terms of the sequence to confirm it is correct
6. If the explicit formula does not match the initial terms, try to find a new pattern or check for errors in the previous steps.

Example: Find the general formula for the recursive sequence defined by the following:

a1 = 2 an = 3an-1 – 2

Step 1: Identify the first few terms of the sequence: a1 = 2 a2 = 3(a1) – 2 = 3(2) – 2 = 4 a3 = 3(a2) – 2 = 3(4) – 2 = 10 a4 = 3(a3) – 2 = 3(10) – 2 = 28

Step 2: Look for a pattern: We can see that the nth term is equal to 3 times the (n-1)th term minus 2.

Step 3: Express the nth term in terms of the previous terms: an = 3an-1 – 2

Step 4: Solve for the explicit formula using the recursive formula and initial conditions: We know that a1 = 2, so we can substitute that into the recursive formula: an = 3an-1 – 2 a1 = 2 a2 = 3(a1) – 2 = 3(2) – 2 = 4 a3 = 3(a2) – 2 = 3(4) – 2 = 10 a4 = 3(a3) – 2 = 3(10) – 2 = 28

From this, we can see that the explicit formula is: an = 3^n-1 * 2

Step 5: Test the explicit formula using the first few terms of the sequence: a1 = 2 = 3^(1-1) * 2 a2 = 4 = 3^(2-1) * 2 a3 = 10 = 3^(3-1) * 2 a4 = 28 = 3^(4-1) * 2

The explicit formula matches the initial terms of the sequence, so it is correct.