Problem 20 - College Algebra

Question

For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{7}=21\) and \(a_{15}=42 .\)

Step-by-Step Solution

Answer

The first term, \( a_1 \), is 5.25.

Step 1: Identify the formula for an arithmetic sequence

The general formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \), 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: Set up equations for the given terms

Create equations using the nth term formula for the terms we know: \( a_7 = a_1 + 6d = 21 \) and \( a_{15} = a_1 + 14d = 42 \).

Step 3: Subtract the equations to find the common difference

Subtract the first equation from the second: \( (a_1 + 14d) - (a_1 + 6d) = 42 - 21 \). This simplifies to \( 8d = 21 \), giving us \( d = \frac{21}{8} \).

Step 4: Solve for the first term

Use the value of \( d \) in one of the original equations to find \( a_1 \). Using \( a_7 = a_1 + 6d = 21 \), we substitute \( d = \frac{21}{8} \) to get \( a_1 + 6 \cdot \frac{21}{8} = 21 \). Simplifying \( 6 \cdot \frac{21}{8} = \frac{126}{8} = 15.75 \), giving \( a_1 + 15.75 = 21 \) so \( a_1 = 21 - 15.75 = 5.25 \).

Key Concepts

Common Difference

In an arithmetic sequence, the "common difference" is a key aspect. It refers to the consistent amount added to each term to get to the next one. To put it simply, it is the difference between any two consecutive terms. Let's explore this concept with an example.
Consider a sequence: 2, 5, 8, 11. Here, the difference between each pair of consecutive terms (5 - 2, 8 - 5, 11 - 8) is 3. This means the common difference is 3.

In mathematical problems, identifying the common difference (\(d\)) can help in finding other terms of the sequence. In the given exercise, we calculated the common difference using two known terms, \(a_7 = 21\) and \(a_{15} = 42\). By setting up equations from these terms and then subtracting one equation from the other, we determined \(d = \frac{21}{8}\). This method is a classic approach and comes in handy when dealing with arithmetic sequences.

First Term

The "first term" of an arithmetic sequence, often denoted as \(a_1\), is the starting point of the sequence. Knowing \(a_1\) allows you to describe the entire sequence, especially when combined with the common difference.
To find \(a_1\), especially when other terms in the sequence are known, we use the relationship given by the arithmetic sequence formula.
In the step-by-step solution provided, once we found the common difference \(d\), we substituted it into one of the terms' equations. Specifically, for the exercise, we used the equation derived for \(a_7 = 21\), which was \(a_1 + 6d = 21\). Solving for \(a_1\) provided us with the value of 5.25.

Understanding \(a_1\) clarifies the whole arithmetic sequence, as each term can be derived from \(a_1\) by adding multiples of the common difference.

Nth Term Formula

The "nth term formula" is an essential tool in understanding arithmetic sequences. This formula helps to find the value of any term in the sequence without listing all the previous terms. The formula is given as:

\[ a_n = a_1 + (n-1)d \]

Here, \(a_n\) represents the nth term, \(a_1\) is the first term, and \(d\) is the common difference. This compact formula takes advantage of the sequence's constant addition pattern.
When using this formula, you simply plug in the values you know. For example, to find the 15th term \(a_{15}\), if you know \(a_1\) and \(d\), you substitute them into the equation, calculating \(a_1 + 14d = a_{15}\).

This formula is versatile, making it invaluable for arithmetic sequence calculations. It shows how knowing just a couple of terms and the first term intertwines with the whole sequence structure.