Problem 32 - Beginning and Intermediate Algebra

Question

Find the indicated term for each arithmetic sequence. $$a_{1}=10, d=3 ; a_{29}$$

Step-by-Step Solution

Answer

The 29th term of the arithmetic sequence is \(a_{29} = 94\).

Step 1: Write down the given values

We have the values for a₁, d and n as: a₁ = 10 d = 3 n = 29

Step 2: Plug values into the formula

Using the formula for the nth term of an arithmetic sequence, plug in the values. We want to find a₂₉, so substitute n with 29: \[a_{29} = a_{1} + (29-1)d\]

Step 3: Simplify the expression

Begin simplifying the expression by first evaluating the value inside the parentheses: \[a_{29} = a_{1} + (28)d\]

Step 4: Substitute given values

Substitute a₁ = 10 and d = 3: \[a_{29} = 10 + (28)(3)\]

Step 5: Solve for a₂₉

Perform the multiplication and addition to find the value of a₂₉: \[a_{29} = 10 + 84\] \[a_{29} = 94\] Hence, the 29th term of the arithmetic sequence is 94.

Key Concepts

Understanding the nth Term Formula in Arithmetic Sequences

In an arithmetic sequence, each term is derived by adding a fixed number, called the common difference, to the previous term. This simple yet powerful progression can be elegantly captured using the 'nth term formula'. This formula helps you find any term in the sequence without having to list all the terms one by one. The general form of the nth term formula is given by:

\[ a_{n} = a_{1} + (n-1) imes d \]

Where:

• \( a_{n} \) is the nth term you wish to find.
• \( a_{1} \) is the first term of the sequence.
• \( n \) is the term number.
• \( d \) is the common difference between consecutive terms.

To use this, simply identify the values for \( a_{1} \), \( d \), and \( n \), and substitute them into the formula.

Tackling Sequence Problems with Ease

Arithmetic sequence problems can often seem daunting, but they are approachable once you break them down. These problems usually ask for a particular term in the sequence or some properties of the sequence, such as the sum of a few terms.

For instance, if you are tasked to find the 29th term of a sequence where the first term is 10 and the common difference is 3, you can go through these steps:

• Identify and write down what you know: \( a_{1} = 10 \), \( d = 3 \), and you need \( a_{29} \).
• Use the nth term formula to calculate \( a_{29} \).
• Substitute the known values into the formula and simplify step by step.
• Calculate to find the answer. For example, in this problem, the 29th term is found to be 94.

These steps can be applied to any arithmetic sequence problem, making it systematic and simpler to solve.

The Essence of Arithmetic Progression

Arithmetic progression is a foundational concept in algebra and beyond. This type of progression is a sequence of numbers in which each term after the first is obtained by adding a constant (known as the common difference) to the preceding term.

It can be represented as a list of numbers such as 10, 13, 16, 19, ..., where each number (or term) in the sequence is three more than the one before, demonstrating a common difference of 3.

Understanding arithmetic progressions is crucial for grasping more advanced mathematical concepts like series and mathematical induction. This concept highlights the beauty of patterns in mathematics, showing how simple rules can create orderly and predictable sequences from a set of numbers.

• Each arithmetic progression has an easily identifiable pattern due to its constant difference.
• It provides a structured way to analyze linear patterns and trends.
• Learning this concept opens doors to solving real-world counting and arrangement problems.

Mastering arithmetic progressions can enhance problem-solving skills and provide a strong mathematical foundation.