Problem 20 - Algebra 2

Question

Find the 32nd term of each sequence. \(13,17,21,25, \dots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is 137.

Step 1: Identify the common difference

The common difference, \(d\), is found by subtracting any two successive terms in the sequence. So, \(d = 17 - 13 = 4\).

Step 2: Identify the first term of the sequence

The first term of this sequence, denoted \(a_1\), is 13.

Step 3: Substitute the values into the formula for the nth term

Now that we have identified the common difference and the first term, we can substitute these values into the formula for the nth term of an arithmetic sequence. We are supposed to find the 32nd term, so \(n = 32\). Substituting these values into the formula gives: \(a_n = a_1 + (n-1)*d = 13 + (32 - 1)*4 = 13 + 31*4 \).

Step 4: Calculate the 32nd term

To find the 32nd term, we just have to do the multiplication and addition: \(13 + 31*4 = 13 + 124 = 137 \). Thus, the 32nd term of the sequence is 137.

Key Concepts

Common Difference

In an arithmetic sequence, a key characteristic is known as the "common difference." This is the difference between consecutive terms in the sequence. To find it, simply subtract any term from the term that follows it.
In our sequence, the terms are: 13, 17, 21, 25, and so on. By subtracting the first term from the second, we obtain:

• 17 - 13 = 4

This result is consistent when you test it with other consecutive pairs in the sequence:

• 21 - 17 = 4
• 25 - 21 = 4

The common difference remains 4 throughout. Once you have this value, you can easily predict future terms in the sequence by continually adding the common difference.

nth Term Formula

The "nth term formula" for an arithmetic sequence gives you the power to find any term in the sequence without listing all of them. It is represented as: \[ a_n = a_1 + (n-1) \times d \]

• \( a_n \) is the term we want to find
• \( a_1 \) is the first term in the sequence
• \( n \) is the term number
• \( d \) is the common difference

In our sequence, the first term is 13, the common difference is 4, and if, for example, we wanted to find the 32nd term, \( n \) would be 32. That's why we plug in the values as follows: \( a_{32} = 13 + (32-1) \times 4 \). This simplifies to \( 13 + 124 \), yielding the result of 137.

Arithmetic Progression

An "arithmetic progression" is another term for an arithmetic sequence. It refers to a set of numbers in which each term after the first is created by adding a constant known as the "common difference." Here are some important characteristics:

• Predictability: Because each term is derived by adding the common difference to the previous term, the sequence grows in a predictable manner.
• Progression: Arithmetic sequences continue infinitely unless specified otherwise. Their regular intervals make them simple but powerful tools in mathematics.
• Application: Arithmetic progressions are used in various fields such as finance for calculating interest and in computer science for designing algorithms.

In our example of the sequence 13, 17, 21, 25, ..., each number is part of an arithmetic progression created with a common difference of 4. This simple concept allows you to solve complex problems with ease.