Problem 14 - Algebra 2

Question

Find the 32nd term of each sequence. \(0.0023,0.0025,0.0027, \ldots\)

Step-by-Step Solution

Answer

The 32nd term in the sequence is \(0.0085\).

Step 1: Identify the first term (a) and the common difference (d)

The first term (a) is \(0.0023\). The common difference (d) can be found by subtracting the first term from the second term which gives \(0.0025 - 0.0023 = 0.0002\).

Step 2: Use the arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is given as \(a_n = a + (n - 1) * d\). It is known that the first term a is \(0.0023\), the common difference d is \(0.0002\) and want to find the 32nd term so \(n = 32\). Plug these values into the formula.

Step 3: Calculate the 32nd term

Substituting the known values in the formula \(a_n = 0.0023 + (32 - 1) * 0.0002 = 0.0023 + 31 * 0.0002 = 0.0023 + 0.0062 = 0.0085\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the amount that each term increases or decreases compared to the previous one. It's like adding or subtracting the same number over and over to get the next term. To find the common difference, subtract any term from the one that follows it.
For example, in the sequence \(0.0023, 0.0025, 0.0027,\ldots\), we determine the common difference by calculating:

• Subtract the first term from the second term: \(0.0025 - 0.0023 = 0.0002\).

This result, \(0.0002\), is the common difference, meaning each term is obtained by adding \(0.0002\) to the one before it. Understanding the common difference is crucial because it indicates the pattern of change in the sequence.

Nth Term Formula

Arithmetic sequences have a handy tool called the nth term formula, which helps you find any term in the sequence without listing all prior terms. The formula is:

• \(a_n = a + (n - 1) \times d\)

Here, \(a\) is the first term, \(n\) is the term number you want to find, and \(d\) is the common difference.
Using our sequence as an example, find the 32nd term where:

• First term, \(a = 0.0023\)
• Common difference, \(d = 0.0002\)
• Term number, \(n = 32\)

Plug these values into the formula:\[a_{32} = 0.0023 + (32 - 1) \times 0.0002 = 0.0023 + 31 \times 0.0002 = 0.0085\]This formula is perfect for quickly finding any term in an arithmetic sequence.

Arithmetic Progression

An arithmetic progression is simply another name for an arithmetic sequence. It's a sequence of numbers where each term is equal to the previous term, plus (or minus) the common difference. This creates a pattern of consistent change, moving in a straight line when plotted on a graph.
Why is it important? Understanding an arithmetic progression unravel patterns and predict future terms of the sequence with ease.
Reflecting back to our example:

• The sequence starts at \(0.0023\)
• Increases by \(0.0002\)
• Generates a predictable pattern (\(0.0023, 0.0025, 0.0027, \ldots\))

In this context, being able to identify and use arithmetic progressions aids in mathematical modeling and solving real-world problems that require recognition of linear patterns.