Problem 18 - College Algebra with Modeling and Visualization

Question

Use a formula to find the sum of the arithmetic series. $$ 1+3+5+7+\dots+97 $$

Step-by-Step Solution

Answer

The sum of the series is 2401.

Step 1: Identify the Arithmetic Series

First, recognize that the series is arithmetic because each term increases by the same amount. The first term \(a_1 = 1\) and the common difference \(d = 2\).

Step 2: Find the Number of Terms

To find the number of terms \(n\), use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\). We know the last term \(a_n = 97\), so:\[97 = 1 + (n-1) imes 2\]Solving for \(n\):\[96 = (n-1) imes 2\]\[48 = n-1\]\[n = 49\]

Step 3: Apply the Sum Formula

The sum \(S_n\) of an arithmetic series can be found using the formula:\[S_n = \frac{n}{2} imes (a_1 + a_n)\]Substitute \(n = 49\), \(a_1 = 1\), and \(a_n = 97\):\[S_{49} = \frac{49}{2} imes (1 + 97) = \frac{49}{2} imes 98\]

Step 4: Calculate the Sum

Evaluate the expression to find the sum:\[S_{49} = \frac{49}{2} imes 98 = 49 imes 49 = 2401\]Therefore, the sum of the series is 2401.

Key Concepts

Sum of Arithmetic Series

When dealing with the sum of an arithmetic series, you're adding up all the terms in an order where each term consistently increases or decreases by the same amount. What makes an arithmetic series special is the method of finding its sum. If you try to add up each term manually, it can be time-consuming, especially for a long series. However, there is a handy formula to speed up the process, making life much easier.
The formula to find the sum of an arithmetic series is:

• \( S_n = \frac{n}{2} \times (a_1 + a_n) \)

This formula efficiently calculates the total of the series by multiplying the number of terms \( n \) by the average of the first term \( a_1 \) and the last term \( a_n \). By applying this formula, you can find the answer without summing each term individually, which is especially useful for series with many terms!

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a certain pattern, where each number after the first is obtained by adding a constant value. Understanding how to handle arithmetic sequences is crucial for solving problems related to them.
The key formula to find any term in an arithmetic sequence is:

• \( a_n = a_1 + (n-1) \times d \)

This formula lets you calculate any term \( a_n \) in the series by knowing:

• The first term \( a_1 \)
• The common difference \( d \)
• The number of terms \( n \)

If you need to find the last term of a series or verify that a number belongs to the sequence, this formula becomes incredibly handy. Once you have these values, figuring out how many terms are in the series is straightforward, leading directly to finding the sequence's sum.

Common Difference

The common difference in an arithmetic sequence or series is the consistent amount that separates each term. It's a staple concept that defines the sequence's behavior and is consistent throughout.
To identify the common difference \( d \), follow these steps:

• Subtract the first term from the second term: \( d = a_2 - a_1 \).

This should give you the constant difference that is applied from one term to the next. If all calculations are correct, this value stays constant throughout the entire series, signifying each term's consistent increase in value.
Recognizing the common difference is not just vital for understanding the series, but it's also crucial when applying formulas to determine unknown terms or to calculate the sum of the series. In the problem you saw, identifying \( d = 2 \) was critical in both writing the sequence and solving for the number of terms.