Problem 47 - Algebra 2

Question

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=(n+1)^{2} $$

Step-by-Step Solution

Answer

The 4th, 5th, 6th, and 7th terms of the sequence described by the rules are respectively 25, 36, 49, and 64.

Step 1: Find the 4th term

Replace \(n\) with 4 in the general rule \(a_{n} = (n+1)^{2}\). This gives us \(a_{4} = (4+1)^{2} = 5^{2} = 25 \). So, the 4th term of the sequence is 25.

Step 2: Find the 5th term

Next, replace \(n\) with 5 in the general rule. This yields \(a_{5} = (5+1)^{2} = 6^{2} = 36 \). So, the 5th term of the sequence is 36.

Step 3: Find the 6th term

Then, replace \(n\) with 6 in the general rule. This gives us \(a_{6} = (6+1)^{2} = 7^{2} = 49 \). So, the 6th term of the sequence is 49.

Step 4: Find the 7th term

Finally, replace \(n\) with 7 in the general rule. This yields \(a_{7} = (7+1)^{2} = 8^{2} = 64 \). Therefore, the 7th term of the sequence is 64.

Key Concepts

Understanding Algebra in Sequences

In arithmetic sequences, algebra plays a foundational role in finding the terms of a sequence. Algebra helps us express general mathematical ideas easily. For instance, the expression \( a_{n} = (n+1)^{2} \) is an algebraic formula representing a sequence.

• Algebraic expressions: These are combinations of numbers, variables, and arithmetic operations.
• Variables: A symbol, like \(n\) in our case, representing numbers that can vary.
• Expressions: Combining these variables and numbers using operations like addition and exponentiation.

When we use algebra in sequences, we can quickly find the nth term, understand patterns, and solve problems efficiently.

Steps in Term Calculation

Term calculation in sequences involves applying a formula to find specific terms. In our example, to find the 4th, 5th, 6th, and 7th terms, we follow these steps:
Step by Step:

• Identify the term position, \(n\).
• Substitute \(n\) into the given expression \((n+1)^{2}\).
• Perform the arithmetic calculations: addition first, followed by squaring the result.
• Repeat the process for each term position.

For instance, for the 4th term: Substitute 4 into \(a_{n} = (n+1)^{2}\). This results in \(5^2\) after doing the computations, giving the 4th term as 25.
Applying these steps allows for precise calculation of any term in the sequence.

Using Sequence Formulas

Sequence formulas are crucial for defining and calculating terms within a sequence. These formulas take the form of an equation that captures the pattern of the sequence.

• A formula gives us a rule to find any term without having to list all previous terms.
• For example, \((n+1)^{2}\) is a sequence formula that generates the terms based on the position \(n\).
• Each term in a sequence is determined by substituting \(n\) with the term number in the formula.

The power of sequence formulas lies in their ability to simplify calculations and predictions about a sequence.

The Role of Power Functions

Power functions involve expressions where numbers are raised to the power of a variable or another number. In our sequence, the formula \((n+1)^{2}\) is a power function, with \(n+1\) raised to the power of 2.

• Exponentiation: In our formula, for any given \(n\), we add one and raise it to the power of two.
• Quadratic Nature: Since it's squared, the sequence follows a quadratic pattern, meaning each term grows at an increasing rate.
• Curiosity and Exploration: Understanding power functions helps us see why the sequence values increase rapidly. It highlights the relation between algebra and real-world exponential growth scenarios.

Grasping power functions enables us to predict and understand the behavior of various mathematical sequences and models.