Problem 20 - College Algebra

Question

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{150}\) when \(a_{1}=-60, d=5\)

Step-by-Step Solution

Answer

The 150th term of the arithmetic sequence is 685.

Step 1: Identify the given information

The first term, \( a_{1} = -60 \) and the common difference, \( d = 5 \). What needs to be found is the 150th term, which can be represented by \( a_{150} \).

Step 2: Apply the Arithmetic sequence formula

The formula for the nth term in an arithmetic sequence is \( a_{n} = a_{1} + (n - 1)d \). Substituting the given information into the formula, the expression will be \( a_{150} = -60 + (150 - 1) * 5 \).

Step 3: Solve the equation

Now solve the expression: \( a_{150} = -60 + 149 * 5 = -60 + 745 = 685 \).

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a specific pattern where each term after the first is obtained by adding a constant value, known as the 'common difference', to the preceding term. To find any term in an arithmetic sequence, we use the arithmetic sequence formula, which is expressed as:
\[ a_n = a_1 + (n - 1)d \]
where \( a_n \) represents the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number. This formula allows us to calculate the value of any term in the sequence directly without having to write out the entire sequence. For students working on homework problems, understanding this formula is key to finding terms in an arithmetic sequence efficiently.

Common Difference in Sequences

The 'common difference' in an arithmetic sequence is the consistent interval or difference between consecutive terms. It's the amount by which each term increases (or decreases, if the common difference is negative) to reach the next term in the sequence. To identify the common difference, you can subtract any term from the one following it:
\[ d = a_{n+1} - a_n \]
In our example from the textbook, the common difference is \( d = 5 \), which means each term in the sequence is 5 more than the term before it. The concept of a common difference is fundamental to understanding arithmetic sequences because this value remains constant throughout the sequence and governs its progression.

Sequence Terms Calculation

When calculating terms in an arithmetic sequence, we don't need to list all the elements to reach the one we're interested in. Instead, we can apply the arithmetic sequence formula directly to find any term. For instance, to find the 150th term of the sequence with a given first term and common difference, we substitute the appropriate values into the arithmetic sequence formula:
\[ a_{150} = a_1 + (150 - 1)d \]
This direct approach bypasses the need for repetitive and time-consuming addition. It's a more efficient method, especially when dealing with large term numbers like in our exercise where we sought the 150th term. Sequence terms calculation utilizing the arithmetic sequence formula saves time and reduces the chances of making errors that can occur when manually counting or adding up a long list of numbers.