# How to Write a Formula for a Recursive Sequence

A sequence in which the value of each statement is determined by the previous statement in the sequence is called 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:

- Start by identifying the first few terms of the recursive sequence.
- Attempt to find a pattern in the terms, such as a specific difference or ratio between consecutive terms.
- Use the pattern identified in step 2 to express the nth term of the sequence in terms of the previous terms.
- Solve for the explicit formula using the recursive formula found in step 3 and the initial conditions of the sequence.
- Test the explicit formula using the first few terms of the sequence to confirm it is correct
- 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.

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