Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Use a formula to find the sum of each arithmetic series. $$1+2+3+4+\dots+50$$

Step-by-Step Solution

Answer

The sum of the series is 1275.

Step 1: Identify the First Term (a) and the Last Term (l)

In the arithmetic series given, the first term is 1, and the last term is 50.

Step 2: Determine the Number of Terms (n)

To find the number of terms, note that the series starts at 1 and ends at 50. Thus, the number of terms is simply 50, since each number from 1 to 50 is included.

Step 3: Use the Formula for the Sum of an Arithmetic Series

The formula to find the sum of an arithmetic series is \( S_n = \frac{n}{2} \times (a + l) \), where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. Plug in the values: \( S_{50} = \frac{50}{2} \times (1 + 50) \).

Step 4: Calculate the Sum

Perform the calculations: \( \frac{50}{2} = 25 \) and \( 1 + 50 = 51 \). Thus, \( S_{50} = 25 \times 51 = 1275 \).

Key Concepts

Sum Formula

An arithmetic series is a sequence of numbers in which the difference between any two successive terms is constant. To find the sum of an arithmetic series, we use a straightforward formula. This formula helps us avoid the tedious work of adding up each term individually. It is given by: \[S_n = \frac{n}{2} \times (a + l)\]Where:

• \( S_n \) is the sum of the series.
• \( n \) is the number of terms.
• \( a \) is the first term.
• \( l \) is the last term.

This formula significantly simplifies the calculation, as it finds a sum based on the average of the first and last terms of the series, multiplied by the number of terms. This is particularly useful when dealing with a large number of terms, saving both time and effort.
Try applying this formula to practice deriving sums quickly and efficiently.

Number of Terms

Determining the number of terms in an arithmetic series is crucial for using the sum formula. You need to know how many numbers are in the sequence from the first to the last term. For straightforward sequences where each term increases or decreases by 1, simply subtract the first term from the last term and add 1. For example, in the series from 1 to 50, the calculation is:\[50 - 1 + 1 = 50\]Remember that if the common difference is not 1, you will have to consider it in your calculations. However, in linear sequences starting from 1, this method works directly.
Mastering this basic skill is essential as you continue to work through arithmetic problems and can help simplify your calculations greatly.

First and Last Terms

In any arithmetic series, identifying the first and last terms is the starting point. These terms define the scope of your series and are essential inputs for the sum formula. The first term, often denoted as \( a \), is the beginning of your sequence. It establishes the point from which all subsequent terms are based. The last term, denoted as \( l \), marks the endpoint of the sequence.Consider our example:

• The first term \( a \) is 1.
• The last term \( l \) is 50.

These terms help you determine both the entirety of the series and anchor the sum formula in concrete numbers. Having a clear understanding of how to determine these terms is vital for solving arithmetical series problems accurately.
Always verify your first and last terms before moving on to further calculations.