Problem 46 - Algebra 2

Question

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=91, d=-4, a_{n}=15 $$

Step-by-Step Solution

Answer

The sum of the series is 1060.

Step 1: Understand the formula for the nth term

In an arithmetic series, the formula for the nth term is given by \( a_n = a_1 + (n-1) \cdot d \). Here we need to find the value of \( n \), the total number of terms.

Step 2: Use the nth term formula to find n

We have \( a_1 = 91 \), \( d = -4 \), and \( a_n = 15 \). Substitute these into the formula: \( 15 = 91 + (n-1)(-4) \). Simplifying, we get \( 15 = 91 - 4(n-1) \).

Step 3: Solve the equation for n

Rearrange \( 15 = 91 - 4(n-1) \) to find \( n \):\( 15 = 91 - 4n + 4 \), which simplifies to \( 15 = 95 - 4n \), and then \( 4n = 80 \). Dividing both sides by 4, we find \( n = 20 \).

Step 4: Use the sum formula for n terms

The formula for the sum of the first \( n \) terms of an arithmetic series is \( S_n = \frac{n}{2}(a_1 + a_n) \). We have \( n = 20 \), \( a_1 = 91 \), and \( a_n = 15 \).

Step 5: Substitute values into the sum formula

Insert the values into the sum formula:\[ S_{20} = \frac{20}{2} \times (91 + 15) \]which simplifies to:\[ S_{20} = 10 \times 106 \]Calculating gives \[ S_{20} = 1060 \].

Key Concepts

Understanding the nth Term Formula

The nth term formula is a handy tool when dealing with arithmetic sequences. It's used to find any term in a sequence when you know the first term and the common difference. The formula is:\[ a_n = a_1 + (n-1) \cdot d \]where:

• \( a_n \) is the nth term you want to calculate.
• \( a_1 \) is the first term of the sequence.
• \( n \) represents the term number you're interested in.
• \( d \) is the common difference between consecutive terms.

For example, if your first term is 91, the common difference is -4, and you're trying to find the 15th term, you'd plug these values into the formula to solve for \( n \). This step is crucial before moving on to calculating the sum of the series. Once \( n \) is identified, it serves as a bridge to the sum of the series calculation.

Summing Up with the Sum of the Arithmetic Series

The sum of an arithmetic series helps you find the total of all terms in a sequence up to the nth term. The formula for this sum is as follows:\[ S_n = \frac{n}{2}(a_1 + a_n) \]This formula requires a few known elements:

• \( S_n \) is the sum of the series up to the nth term.
• \( a_1 \) is the first term of the sequence.
• \( a_n \) is the nth or the last term of the sequence.
• \( n \) is the total number of terms.

To use the formula, simply plug in the known values. For example, with \( n=20 \), the first term \( a_1 = 91 \), and the nth term \( a_n = 15 \), you can find:\[ S_{20} = \frac{20}{2}(91 + 15) \]\[ S_{20} = 10 \times 106 \]\[ S_{20} = 1060 \]It quickly gives the total of the series, saving time when summing long sequences manually.

Solving Linear Equations to Find Terms

Linear equations are often part of mathematics problems like finding terms in an arithmetic sequence. Solving a linear equation essentially means finding the value of \( n \) or another variable. Here's how it works using our previous example:
Start with the equation derived from the nth term formula:\( 15 = 91 - 4(n-1) \).

• You first simplify by distributing the -4.
• Combine like terms to set up:\( 15 = 95 - 4n \).
• Isolate the variable by moving all terms involving \( n \) to one side:\( 4n = 80 \).
• Finally, divide through to solve for \( n \):\( n = \frac{80}{4} = 20 \).

This process illustrates how arithmetic and algebra merge to find how many terms are needed to reach a certain value in a series. Understanding and practicing these steps can enhance problem-solving skills significantly.