Problem 46 - Precalculus: Mathematics for Calculus

Question

Find the indicated term of the arithmetic sequence with the given description. The 100 th term is \(-750\), and the common difference is \(-20\). Find the fifth term.

Step-by-Step Solution

Answer

The fifth term of the sequence is 1150.

Step 1: Identify the Formula

In an arithmetic sequence, the nth term can be found using 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 2: Establish Known Values

We know the 100th term \( a_{100} = -750 \) and the common difference \( d = -20 \). We need to find the fifth term \( a_5 \).

Step 3: Find the First Term

Using the formula \( a_n = a_1 + (n-1)d \) with \( n = 100 \), we set up the equation: \( -750 = a_1 + (100-1)(-20) \). Simplify to: \( -750 = a_1 - 1980 \).

Step 4: Solve for the First Term

Rearrange the equation \( -750 = a_1 - 1980 \) to \( a_1 = -750 + 1980 \). Calculate \( a_1 = 1230 \).

Step 5: Find the Fifth Term

Now that we have \( a_1 \), we can find the fifth term using \( a_5 = a_1 + (5-1)d \). Substitute the values: \( a_5 = 1230 + 4(-20) \).

Step 6: Calculate the Fifth Term

Simplify the expression: \( a_5 = 1230 - 80 \). Hence, \( a_5 = 1150 \).

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. This difference is crucial for understanding how the sequence progresses. To find any term in an arithmetic sequence, we use a straightforward formula:

• \( a_n = a_1 + (n-1)d \)

In this equation:

• \( a_n \) represents the nth term of the sequence that you want to find.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference, which we'll explain more soon.
• \( n \) is the term number you are interested in.

Understanding this formula helps you predict any term in the sequence without writing out every previous term. Once you know the initial term and the common difference, finding any term is just a matter of simple arithmetic.

Common Difference

The common difference, symbolized as \( d \) in the arithmetic sequence formula, is simply the amount we add (or subtract, if it's negative) to each term to get to the next term. It defines how quickly the terms are changing.

• If \( d \) is positive, the sequence is increasing.
• If \( d \) is negative, like in our problem, the sequence is decreasing.

To find the common difference, you can subtract any term from the next term. For the sequence in our problem, the common difference is given as \(-20\). This tells us that each term is 20 less than the previous one, which is why the sequence is decreasing. Knowing the common difference is key to understanding the pattern and predicting other terms in the sequence.

Nth Term

The nth term, \( a_n \), is the term we aim to find—it represents any term in an arithmetic sequence based on its position, \( n \). Knowing how to calculate the nth term allows us to dive into any part of the sequence without calculating each preceding term.In our problem, we needed the fifth term, \( a_5 \). Using the arithmetic sequence formula, we substituted \( n \) with 5. This formula leveraged our known values for the first term, \( a_1 \), and the common difference, \( d \), allowing us to find the fifth term where \( a_5 = a_1 + 4d \).With these elements, finding any term in a sequence is a quick calculation once you understand the sequence's dynamics. No need for lengthy computations, simply apply the formula and solve for your desired \( n \).