Problem 34 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the first 20 terms for the arithmetic sequence. $$ a_{8}=4, a_{10}=14 $$

Step-by-Step Solution

Answer

The sum of the first 20 terms is 330.

Step 1: Understand the Arithmetic Sequence Formula

The general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Step 2: Set Up Equations for Given Terms

We have \( a_8 = 4 \) and \( a_{10} = 14 \). Using the formula, we set up the following equations:1. \( a_8 = a_1 + 7d = 4 \)2. \( a_{10} = a_1 + 9d = 14 \)

Step 3: Solve for the Common Difference and First Term

To find \( d \), subtract the first equation from the second:\((a_1 + 9d) - (a_1 + 7d) = 14 - 4\)This simplifies to:\(2d = 10\)Solving for \( d \), we get \( d = 5 \).Substitute \( d = 5 \) into the first equation:\( a_1 + 7(5) = 4 \)\( a_1 + 35 = 4 \)\( a_1 = -31 \).

Step 4: Use the Formula for the Sum of an Arithmetic Sequence

The sum of the first \( n \) terms of an arithmetic sequence is given by the formula:\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]For the first 20 terms, \( n = 20 \), \( a_1 = -31 \), and \( d = 5 \).

Step 5: Calculate the Sum of the First 20 Terms

Substitute the known values into the sum formula:\[ S_{20} = \frac{20}{2} (2(-31) + (20-1)5) \]\[ S_{20} = 10(-62 + 95) \]\[ S_{20} = 10 \times 33 \]\[ S_{20} = 330 \]

Key Concepts

Arithmetic Series Formula

To find the sum of an arithmetic series, we use a special formula. This formula helps us add up all the terms in the sequence quickly and easily. The formula to calculate the sum of the first \( n \) terms is:\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]Here’s what each part represents:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) is the number of terms you are adding together.
• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between terms.

This formula is very handy because it combines all necessary elements to calculate the total of a large sequence effectively. Instead of adding each term one by one, you use the properties of arithmetic sequences to simplify the process.

Common Difference

The common difference, denoted as \( d \), is a key component of an arithmetic sequence. This is the constant amount that each term increases (or decreases) from the previous one.In every arithmetic sequence:

• The common difference is found by subtracting any term from the following term.
• If the sequence is increasing, \( d \) is positive.
• If the sequence is decreasing, \( d \) is negative.

Calculating \( d \) is crucial because it tells us how the sequence progresses. In our specific example, where \( a_8 = 4 \) and \( a_{10} = 14 \), we found \( d \) by using the equation \( 2d = 10 \). Solving gives us \( d = 5 \), showing that each term increases by 5.

Nth Term Formula

To find any term in an arithmetic sequence, we use the nth term formula. This formula helps you determine the value of a particular term without listing all previous terms.The formula for the nth term is:\[ a_n = a_1 + (n-1)d \]Where:

• \( a_n \) is the value of the nth term.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference.

In problem-solving, this formula simplifies finding specific terms greatly. For instance, if \( a_1 = -31 \) and \( d = 5 \), you can quickly find any other term, such as \( a_8 \) or \( a_{10} \), by plugging in these values into the formula.

Sum of Arithmetic Series

Finding the sum of an arithmetic series involves combining all elements of the sequence up to a certain term. Using the formula for the sum of the first \( n \) terms, we calculate the total efficiently:\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]Let's break it down for the problem at hand:

• First, find the first term \( a_1 \), which we calculated as -31.
• Determine \( d \), the common difference, calculated as 5.
• Choose \( n \), the number of terms, in this case, 20.

So, for the first 20 terms, insert these values into the formula:\[ S_{20} = \frac{20}{2} (2(-31) + (20-1)5) \]Simplify it to get the sum \( S_{20} = 330 \). This calculation presents how formulas simplify complex arithmetic operations by consolidating multiple steps into a single mathematical expression.