Problem 57 - Algebra 2

Question

The 30 th term of a finite arithmetic series is 4.4 . The sum of the first 30 terms is \(78 .\) What is the first term of the series?

Step-by-Step Solution

Answer

The first term of the series is 23.6

Step 1: Write down the formulas

Let's write down the formulas for the nth term and the sum of an arithmetic series: \(a_n = a + (n-1)d\) and \(S = 1/2 n [2a + (n-1)d]\)

Step 2: Apply the properties

We know from the problem that \(a_{30} = 4.4\) and \(S_{30} = 78\). Substituting these values into the formulas from Step 1, we get the following two equations: \(4.4 = a + 29d\) and \(78 = 1/2 * 30 [2a + 29d]\)

Step 3: Solve the system of equations

Solving this system of equations can lead us to find the values of \(a\) and \(d\). Double the first equation to eliminate \(d\) in the second equation: \(8.8 = 2a + 58d\). Subtract this from the second equation to get an equation in terms of \(a\): \(47.2 = 2a\), which simplifies to \(a = 23.6\).

Key Concepts

nth Term

In arithmetic series, the "nth term" is a specific element in a sequence. To find the nth term of an arithmetic sequence, we apply the formula:

• \( a_n = a + (n-1)d \)

This formula is useful for pinpointing any term's value based on its position.
The variable "a" is the first term of the series. "d" stands for the common difference, which is the amount you add to each term to get the next one.
The formula allows us to express each term using the start of the series and the step size, which is crucial for solving problems related to the position of terms in a series.

Sum of Arithmetic Series

When we talk about the sum of an arithmetic series, we refer to the total value when adding all terms from the beginning up to a specified term. This sum is calculated using:

• \( S = \frac{n}{2} \times (2a + (n-1)d) \)

Here, "S" represents the sum of the series up to "n" terms.
"n" is the number of terms being added. The term \((2a + (n-1)d)\) accounts for the series expanding from the first term.
This formula helps by condensing potentially long calculations into a straightforward expression, making it easier to find the sum without manually adding each term.

Finite Series

A finite series in mathematics refers to a series with a definite number of terms. In the context of arithmetic series, this means you're working with a sequence that has a clear beginning and an end.

• Finite series are manageable because they involve a set number of terms.

The problem statement in our exercise mentions the 30-term arithmetic series as finite, indicating we are looking at just 30 elements.
By understanding the start and the end of the series, along with how terms are spaced (the common difference), we can calculate not only individual terms but also make broader calculations about the series, such as finding the first term or the overall sum.
Recognizing a series as finite allows for practical usage, especially in solving problems where the exact count of terms is essential.