Problem 11 - Algebra 2

Question

Find the 32nd term of each sequence. \(34,37,40,43, \ldots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is 127.

Step 1: Identify sequence type and properties

Here the increase of 3 between the consecutive terms indicates it's an arithmetic sequence, with first term (a1) = 34 and common difference (d) = 3.

Step 2: Plug into formula

The formula for the nth term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substituting the values yields: \(a_{32} = 34 + (32 - 1) * 3\).

Step 3: Perform Calculation

To find \(a_{32}\), perform the operations: \(a_{32} = 34 + 93 = 127\).

Key Concepts

nth term formula

The nth term formula is a keystone in understanding arithmetic sequences. In an arithmetic sequence, the sequence is formed by repeatedly adding a fixed number, which is known as the "common difference," to the previous term. This formula helps to directly find any term in the sequence without listing all preceding terms. To use this formula, you need to know:

• The first term of the sequence, denoted as \(a_1\).
• The common difference, which is the difference between consecutive terms, denoted as \(d\).

The formula itself is expressed as: \[a_n = a_1 + (n - 1)d\]In this equation:

• \(a_n\) is the term you want to find.
• \(n\) represents the position of the term in the sequence.

Existing only within the realm of arithmetic sequences, this formula makes it easy to calculate, for instance, the 1000th term, by substituting the known values into the formula. It saves time and ensures efficiency when dealing with long sequences.

common difference

The "common difference" is a fundamental aspect of an arithmetic sequence. It is the constant value added to each term to get the next one. This difference is what gives arithmetic sequences their unique linear structure. To calculate the common difference (\(d\)), you subtract a term from the term that follows it. For example, let's use the sequence given in the exercise: \(34, 37, 40, 43, \ldots\).From one term to the next, you can observe:

• \(37 - 34 = 3\)
• \(40 - 37 = 3\)
• \(43 - 40 = 3\)

Since this value (\(d = 3\)) is consistent throughout the sequence, it confirms that we're working with an arithmetic sequence. Understanding this common difference is crucial because it allows you to predict future terms in the sequence reliably. Every arithmetic sequence is defined by its first term and its common difference, making calculations and pattern recognitions straightforward.

arithmetic progression

An arithmetic progression is simply another term for an arithmetic sequence. This progression is an ordered set of numbers where the pattern between them is consistent, defined by the common difference. Consider the sequence in the exercise: \(34, 37, 40, 43, \ldots\). Each term increases by a fixed amount - in this case, by 3. This consistent adding of a fixed number is what forms an arithmetic progression.Key characteristics of arithmetic progressions include:

• The sequence can be increasing or decreasing, depending on whether the common difference is positive or negative.
• The nature of the sequence being linear, meaning the pattern doesn't change.
• The ability to calculate any term using the nth term formula.

Knowing about arithmetic progressions helps in various mathematical calculations, such as finding specific terms, summing terms, and understanding linear trends in data sets. This characteristic of predictability and simplicity makes arithmetic progressions an important concept across different areas of mathematics and real-life applications.