Problem 16 - Beginning and Intermediate Algebra

Question

Given the general term of each sequence, find each of the following. \(a_{n}=3 n-11\) a) the first term of the sequence b) \(a_{7}\) c) the 32 nd term

Step-by-Step Solution

Answer

The first term of the sequence is -8, the seventh term is 10, and the 32nd term is 85.

Step 1: First Term#a_{1}

To find the first term of the sequence, we substitute n = 1 into the general term formula: \(a_{1} = 3(1) - 11\).

Step 2: Calculating First Term

Simplify the expression to find \(a_{1}\): \( a_{1} = 3 - 11 = -8\). So, the first term of the sequence is -8.

Step 3: Seventh Term#a_{7}

To find the seventh term of the sequence, we substitute n = 7 into the general term formula: \(a_{7} = 3(7) - 11\).

Step 4: Calculating Seventh Term

Simplify the expression to find \(a_{7}\): \(a_{7} = 21 - 11 = 10\). So, the seventh term of the sequence is 10.

Step 5: 32nd Term#a_{32}

To find the 32nd term of the sequence, we substitute n = 32 into the general term formula: \(a_{32} = 3(32) - 11\).

Step 6: Calculating 32nd Term

Simplify the expression to find \(a_{32}\): \(a_{32} = 96 - 11 = 85\). So, the 32nd term of the sequence is 85. In summary, the first term of the sequence is -8, the seventh term is 10, and the 32nd term is 85.

Key Concepts

General Term Formula

Arithmetic sequences follow a specific pattern characterized by equal differences between consecutive terms. To describe any term in such a sequence, we use the **General Term Formula**. The purpose of this formula is to allow us to find terms at specific positions within the sequence without having to list out all previous terms.
The general term formula for an arithmetic sequence is expressed as:\[ a_n = a_1 + (n-1)d \]In this formula:- \(a_n\) represents the term at the nth position.- \(a_1\) is the first term of the sequence.- \(d\) stands for the common difference between terms.- \(n\) is the term number we're trying to find.
By substituting the known values for starting term, common difference, and desired term position, we can easily determine any term within the sequence.

Finding Terms in a Sequence

Finding specific terms in an arithmetic sequence involves substituting the term position into the general term formula. This exercise asks us to determine the first, the seventh, and the thirty-second terms of a sequence given by the formula \(a_{n} = 3n - 11\).

• To find the first term, substitute \(n = 1\) into the equation. This gives \(a_{1} = 3(1) - 11\).
• For the seventh term, substitute \(n = 7\) to find \(a_{7} = 3(7) - 11\).
• To locate the thirty-second term, you'll use \(n = 32\) resulting in \(a_{32} = 3(32) - 11\).

By plugging these values into the formula, we can efficiently compute any term in the sequence without having to sequentially add to reach the desired position.

Substitution in Algebra

**Substitution in Algebra** is a key technique used when working with equations and formulas. It involves replacing a variable with a number or another expression to simplify or evaluate the equation.
In our exercise, we perform substitution when we replace \(n\) with specific numbers such as 1, 7, and 32. This step is crucial for finding particular terms within an arithmetic sequence.
For instance, substituting \(n = 1\) in the general term formula \(a_{n} = 3n - 11\) means you'll compute \(3 \times 1 - 11\), streamlining the process to solve for the first term directly. Similarly, substituting other values helps derive specific terms like \(a_{7}\) and \(a_{32}\) with ease.

Evaluating Algebraic Expressions

Once substitution is complete, the next step usually involves **Evaluating Algebraic Expressions**. This means simplifying the expression to find the numeric value of a specific term.
The process of evaluation here involves carrying out basic arithmetic operations after substitution. For example:- When you find \(a_{1} = 3(1) - 11\), simplifying yields \(3 - 11 = -8\).- Similarly, evaluating \(a_{7} = 21 - 11\) gives you 10.- Lastly, evaluating \(a_{32} = 96 - 11\) results in 85.
These evaluations are straightforward once substitution has been done. Ensure each step follows order of operations (PEMDAS/BODMAS) to arrive at the correct result. By mastering these evaluations, you can solve for various terms effortlessly.