Problem 16 - Algebra 2

Question

Find the 32nd term of each sequence. \(213,201,189,177, \ldots\)

Step-by-Step Solution

Answer

The 32nd term of the sequence is -159.

Step 1: Identify the first term and the common difference

The first term \(a\) is 213. The common difference \(d\) can be found by subtracting the second term from the first term, or the third term from the second, and so on. When we subtract 201 from 213, or 189 from 201, we get -12. So, \(d\) is -12.

Step 2: Apply the formula for the nth term of an arithmetic progression

The formula for the nth term of an arithmetic progression is \(a + (n-1) * d\). Substituting \(a = 213\), \(d = -12\), and \(n = 32\) into the formula, we get the 32nd term is equal to \(213 + (32-1) * (-12)\)

Step 3: Simplify the expression

Simplify the expression by first calculating the multiplication and then performing the addition, yielding \(213 + 31 * (-12) = 213 - 372\). Since subtracting a larger number from a smaller one gives a negative result, \(213 - 372 = -159\)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the consistent amount that separates each pair of consecutive terms. It is the key feature of an arithmetic progression. You can find this value by subtracting any term from the term that follows it.

In the sequence given in the exercise, starting with 213, we subtract 201 from 213 and end up with a difference of \(-12\). Similarly, subtracting 189 from 201 once again results in \(-12\). This tells us that the common difference for this sequence is \(-12\), meaning that each term decreases by 12 from the previous term.

Identifying the common difference is crucial for determining any term within the sequence, whether it is the next one, the 10th, or the 32nd, as in this exercise.

Arithmetic Progression Formula

The arithmetic progression formula is a mathematical tool used to determine any term in a sequence when you know the first term and the common difference. The general nth term formula for an arithmetic sequence is:
\[a_n = a + (n-1) \times d\]
- \(a_n\) denotes the nth term.
- \(a\) represents the first term of the sequence.
- \(d\) is the common difference between terms.
- \(n\) is the specific term number you are interested in.

For instance, in our example, if we wish to find the 32nd term, we substitute the values into the formula:
\(a = 213\), \(d = -12\), and \(n = 32\). This would look like:
\[a_{32} = 213 + (32-1) \times (-12)\]
This formula simplifies the process of finding any term in an arithmetic sequence efficiently.

Nth Term of a Sequence

Finding the nth term of a sequence involves applying the arithmetic progression formula. In our problem, the goal is to identify the 32nd term. Substituting the known values into the formula allows us to perform this calculation easily.

Starting with the formula:
\[a_n = a + (n-1) \times d\]
we substitute to find the 32nd term:
\[a_{32} = 213 + 31 \times (-12)\]
This operations step-by-step proceeds by calculating the multiplication first:
31 multiplied by \(-12\) gives \(-372\). Then, 213 added to \(-372\) gives the result:
\(a_{32} = -159\).

The process demonstrates the utility of the formula, showing how straightforward it is to derive any term from an arithmetic sequence without writing out all the preceding terms.