Problem 18 - College Algebra

Question

Write the first four terms of each sequence. $$a_{1}=5 \text { and } a_{n}=3 a_{n-1}-1 \text { for } n \geq 2$$

Step-by-Step Solution

Answer

The first four terms of the sequence are 5, 14, 41, 122.

Step 1: Identify the First Term

We have the first term given, which is \(a_{1} = 5\).

Step 2: Find the Second Term

We will now find the second term \(a_{2}\)by substituting \(n=2\) into the recursive formula: \(a_{2} = 3a_{2-1} - 1 = 3a_{1} - 1 = 3*5 - 1 = 14\).

Step 3: Find the Third Term

We will now find the third term \(a_{3}\) by substituting \(n=3\) into the recursive formula: \(a_{3} = 3a_{3-1} - 1 = 3a_{2} - 1 = 3*14 - 1 = 41\).

Step 4: Find the Fourth Term

We will now find the fourth term \(a_{4}\) by substituting \(n=4\) into the recursive formula: \(a_{4} = 3a_{4-1} - 1 = 3a_{3} - 1 = 3*41 - 1 = 122\).

Key Concepts

First Terms of a Sequence

When learning about sequences, understanding the initial terms is essential. It's like building a house: you need a solid foundation. In the given problem, the first term of the sequence is provided as \(a_{1} = 5\). This establishes the starting point from which all other terms are calculated.
Each term in a sequence is often dependent on the ones before it, especially in recursive sequences. Knowing the first term allows us to apply any recursive methods or formulas listed to find subsequent terms. It’s crucial to correctly identify and substitute this initial term, because any mistake here can lead to errors in the entire sequence calculation.

Recursive Formula

A recursive formula is a way to express terms of a sequence using previous terms. In simpler terms, it's a type of formula that tells you how to find the next term using the term before it. For the sequence in our problem, the recursive formula is \(a_{n} = 3a_{n-1} - 1\), for \(n \geq 2\).
This specific formula suggests a pattern where you multiply the previous term by 3 and then subtract 1 to find the next term. Recursive formulas are powerful because they succinctly describe potentially complex sequences, allowing you to compute large terms without extensive repetitive calculations.
However, they depend heavily on knowing initial terms, as each step builds directly on the last.

Sequence Calculation

Sequence calculation involves determining specific terms using available information and formulas. With recursive sequences, you start from a known term and repeatedly apply the recursive formula.

**Example Calculation:**

• For the second term \(a_{2}\), the calculation is based on \(a_{1}\): \(a_{2} = 3 \times 5 - 1 = 14\).
• To find the third term \(a_{3}\), use the second term: \(a_{3} = 3 \times 14 - 1 = 41\).
• The fourth term \(a_{4}\) then uses the third term: \(a_{4} = 3 \times 41 - 1 = 122\).

In each step, substitute the previous term into the recursive formula. This method highlights the logical flow and connectivity of sequence terms, illustrating how each term depends on the others.