Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the partial sum without using a graphing utility. $$\sum_{n=1}^{50} n$$

Step-by-Step Solution

Answer

The partial sum of the first 50 natural numbers is 1275.

Step 1: Understand the task

We need to find the partial sum of the first 50 natural numbers. This is an arithmetic series where the first term (a) is 1, the last term (l) is 50 and the number of terms (n) is 50.

Step 2: Apply the formula

The formula for the sum of an arithmetic series is \( S = \frac{n}{2}*(a + l) \). Here, \(a = 1\), \(l = 50\), and \(n = 50\).

Step 3: Calculate the Sum

Substitute the values into the formula to get the sum: \( S = \frac{50}{2}*(1 + 50) \) which simplifies to \( S = 25*(51) = 1275\).

Key Concepts

Partial Sum

A partial sum refers to the sum of a specific part of a sequence or series, rather than the entire series. Think of it as adding up a sequence but stopping at a certain point. For example, if we want the partial sum of natural numbers up to 50, we only add numbers from 1 to 50, rather than continuing indefinitely.
Partial sums are crucial in arithmetic series because they allow us to find the sum of a sequence up to a specific term without listing all the numbers. This makes solving problems more efficient.

• Focus: Specific part of the series
• Example: Sum from 1 to 50
• Benefit: Simplifies complex calculations

Natural Numbers

Natural numbers are the set of positive integers beginning from 1 and moving upwards without end: 1, 2, 3, and so on. They are fundamental in mathematics and are used for counting and ordering.
Natural numbers include just the positive numbers that you would naturally use to count objects. They do not include zero, fractions, or negative numbers.

• Sequence: 1, 2, 3, 4, ...
• Role in Arithmetic: Used in basic operations
• Properties: Infinite and positive

Formula for Sum of Arithmetic Series

The formula for the sum of an arithmetic series is a direct way to find the sum of numbers evenly spaced out in a sequence. An arithmetic series has terms that increase or decrease by the same amount, known as the common difference.
For a series, the sum can be found using: \[ S = \frac{n}{2}(a + l) \] where:

• \( n \): Number of terms
• \( a \): First term
• \( l \): Last term

This formula helps in quickly finding the total of any arithmetic sequence without having to add each term individually. It works effectively for sequences big or small. For instance, this formula was used to find the sum of the first 50 natural numbers, where the result was 1275.