Problem 13 - Algebra 2

Question

Find the 32nd term of each sequence. \(0.1,0.5,0.9,1.3, \dots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is 12.5.

Step 1: Identify the Common Difference

Determine the common difference(d) of the arithmetic sequence by subtracting the first term from the second term. For the given sequence, it should be \(0.5 - 0.1 = 0.4\)

Step 2: Apply the Arithmetic Sequence Formulas

To find the nth term of an arithmetic sequence, use the formula \(a_n = a_1 + (n-1)*d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number and \(d\) is the common difference.

Step 3: Calculate the 32nd Term

Substitute the values into the equation: \(a_{32} = 0.1 + (32 - 1)*0.4 = 0.1 + 31*0.4 = 0.1 + 12.4 = 12.5\) So, the 32nd term of the sequence is 12.5

Key Concepts

Common Difference

The common difference is a crucial concept in understanding arithmetic sequences. This is the fixed amount that is added (or subtracted) from one term to the next to maintain the sequence's uniformity. In an arithmetic sequence, each term increases or decreases by this same value every time. To identify the common difference, one simply subtracts any term from the subsequent term in the sequence.

• For instance, take the sequence: 0.1, 0.5, 0.9, 1.3, ...
• The common difference is calculated as follows: 0.5 - 0.1 = 0.4

It is critical to ensure any sequence examined follows this pattern across all consecutive terms. This consistency confirms the nature of the sequence as arithmetic.

Nth Term Formula

The nth term formula is a powerful tool for determining any term in an arithmetic sequence without listing all the preceding terms. By using this formula, you can jump straight to the desired term.
The formula is given by: \[ a_n = a_1 + (n-1) \times d \]

• Where \(a_n\) is the term you want to find,
• \(a_1\) is the first term of the sequence,
• \(n\) is the term number, and
• \(d\) is the common difference.

Plugging these values into the formula allows you to efficiently solve for any term.
It's particularly useful for large term numbers, like finding the 32nd term in our given sequence.

Sequence Pattern

Recognizing patterns in a sequence is essential in understanding how arithmetic sequences progress. A sequence pattern showcases the uniform manner in which terms increase or decrease. For arithmetic sequences, this pattern is driven by the common difference.
For example, in the sequence 0.1, 0.5, 0.9, 1.3, ... each term increases by 0.4, highlighting the arithmetic pattern.

• Spotting these patterns helps predict future terms.
• It assures that our calculation and formula are applied correctly.

By understanding the pattern, you develop the ability to create sequences or validate the terms already calculated. It's a fundamental step in mastering arithmetic sequences.