Problem 58 - Algebra 2

Question

What is the difference between the third term in the sequence whose recursive formula is \(a_{1}=-5, a_{n}=2 a_{n-1}+1\) and the third term in the sequence whose recursive formula is \(a_{1}=-3, a_{n}=-a_{n-1}+3 ?\) $$ \begin{array}{lllll}{\text { A. } 2} & {\text { B. } 14} & {\text { C. } 20} & {\text { D. } 32}\end{array} $$

Step-by-Step Solution

Answer

The difference between the third term in the first sequence and the third term in the second sequence is -14. So the correct option is B. -14.

Step 1: Calculate the third term of the first sequence

Using the recursive definition for the first sequence, calculate \(a_{3}\). So, to find \(a_{2}\), use the formula to get \(a_{2} = 2(-5) + 1 = -9\). Then, use \(a_{2}\) to get \(a_{3} = 2(-9) + 1 = -17.\)

Step 2: Calculate the third term of the second sequence

Similarly, using the recursive definition for the second sequence, calculate \(a_{3}\). For \(a_{2}=-(-3)+3=6\), then, use \(a_{2}\) to calculate \(a_{3}=-(6)+3=-3.\)

Step 3: Calculate the difference

Now that both third term values, \(a_{3}\), are calculated, their difference can be found by subtracting the third term of the second sequence from the third term of the first sequence, this is \(a_{3, first} - a_{3, second} = -17 -(-3) = -14\).

Key Concepts

Sequence Terms

Sequence terms are individual elements in a sequence, which is a list of numbers arranged in a specific order. When we talk about terms in a sequence, we refer to their position, such as the first term, second term, and so on. Each term is represented by a specific formula, often recursive, that allows us to calculate the terms based on one or more of the previous terms.

For example, in the exercise given, each sequence has its set terms determined by a recursive formula. The first sequence begins with an initial term of \(a_1 = -5\) and defines each subsequent term based on the previous one, thus establishing a clear pattern.

Understanding the terms in a sequence is crucial as it allows us to solve problems related to their order, such as finding a specific term or comparing terms between different sequences.

Recursive Formula

A recursive formula is a way of defining the terms in a sequence using the previous terms. Instead of describing each term individually, a recursive formula states the rule for calculating any term based on the one before it. This makes recursive formulas efficient as you only need to know the initial term and the rule to find any term in the sequence.

In the first sequence from the example, the recursive formula is \(a_n = 2a_{n-1} + 1\). This means each term inside the sequence is twice the previous term, plus one. In the second sequence, the recursive formula is \(a_n = -a_{n-1} + 3\), indicating each term is the negative of the previous term plus three.

• Initial value or base, such as \(a_1 = -5\) or \(a_1 = -3\).
• Recursive rule to determine the next term from the previous one.

Recursive formulas are especially useful in algebra and other areas of mathematics as they provide a systematic method for continuing the sequence indefinitely.

Algebra Problems

Algebra problems often involve sequences, and understanding recursive sequences can be crucial for solving these problems. The exercise involves calculating specific sequence terms and then identifying their differences, a common type of algebra problem.

To solve this type of problem, follow these steps:

• Identify the initial term from each sequence.
• Use the recursive formula to calculate subsequent terms, specifically the ones needed for the problem (here, the third term).
• Compare results or differences between terms of different sequences.

The key to mastering these problems is the ability to understand and apply recursive formulas correctly to predict terms. The final step often involves a comparison, requiring basic operations like subtraction, as shown in the provided example where the two sequence terms' difference is calculated.

By breaking down these steps clearly, recursive sequences become manageable, enabling you to confidently tackle algebra problems related to them.