Problem 35 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=2 ; \quad a_{n}=3+a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms are 2, 5, 8, 11, 14.

Step 1: Identify the First Term

The first term in the sequence is given directly in the problem: \(a_1 = 2\). This is our starting point.

Step 2: Determine the Second Term

Using the recursive formula \(a_n = 3 + a_{n-1}\), we can find the second term. Substitute \(a_1\) into the formula: \(a_2 = 3 + a_1 = 3 + 2 = 5\).

Step 3: Calculate the Third Term

To find the third term, use the value of \(a_2\): \(a_3 = 3 + a_2 = 3 + 5 = 8\).

Step 4: Find the Fourth Term

Next, determine the fourth term using \(a_3\): \(a_4 = 3 + a_3 = 3 + 8 = 11\).

Step 5: Compute the Fifth Term

Finally, use \(a_4\) to find the fifth term: \(a_5 = 3 + a_4 = 3 + 11 = 14\).

Key Concepts

First Term

The first term of a recursive sequence is your starting point and is often provided directly. For this problem, it's given that the first term, denoted as \(a_1\), is 2. You need to know this value because it is essential for finding the subsequent terms in the sequence.

In other words, everything starts from \(a_1\). Always make sure you correctly identify this term, as any inaccuracies will affect the entire sequence.

Here, knowing that \(a_1 = 2\) allows us to use this information as a foundation for all the following calculations.

Recursive Formula

A recursive formula allows you to determine the next term in the sequence based on the previous term. In this problem, the recursive formula is \(a_n = 3 + a_{n-1}\). This means that to find any term \(a_n\), you simply take the previous term \(a_{n-1}\) and add 3 to it.

Understanding this concept is crucial because it tells you the pattern of the sequence. You follow this pattern to find each new term.

For example, to find \(a_2\), you apply the formula using \(a_1\):
\(a_2 = 3 + a_1 = 3 + 2 = 5\). Each new term is determined this way, ensuring a consistent approach throughout the sequence.

Sequence Calculation

Once you have the first term \(a_1\) and the recursive formula, you can calculate the rest of the sequence by continually applying the formula. Let's go through the steps given in the solution to compute the first five terms:

- Step 1: Start with \(a_1 = 2\), which is the first term.
- Step 2: Use the recursive formula to find the second term: \(a_2 = 3 + a_1 = 3 + 2 = 5\).
- Step 3: Apply the same formula to find the third term: \(a_3 = 3 + a_2 = 3 + 5 = 8\).
- Step 4: Continue the pattern to get the fourth term: \(a_4 = 3 + a_3 = 3 + 8 = 11\).
- Step 5: Finally, find the fifth term: \(a_5 = 3 + a_4 = 3 + 11 = 14\).

By understanding how to use the first term and the recursive formula, you can extend this method to find as many terms in the sequence as you need. This systematic approach ensures you can generate the sequence accurately.