Problem 39 - College Algebra

Question

Find \(1+2+3+4+\dots+100,\) the sum of the first 100 natural numbers.

Step-by-Step Solution

Answer

The sum of the first 100 natural numbers is 5050.

Step 1: Identify the arithmetic series

Firstly, we identify that the sum \(1+2+3+4+\dots+100\) is an arithmetic series. In this case, the first term \(a\) is 1 and the last term \(l\) is 100.

Step 2: Apply the arithmetic series sum formula

The formula for the sum of an arithmetic series is \(S=\frac{n}{2}[a + l]\). In this case, \(n=100\), \(a=1\), and \(l=100\). We substitute these values into the formula, which gives us \[S = \frac{100}{2}[1 + 100].\]

Step 3: Compute the sum

The arithmetic simplifies the formula to \[S = 50 [101] = 5050\]. So, the sum of the first 100 natural numbers is 5050.

Key Concepts

Sum of Natural Numbers

The sum of natural numbers refers to the total you get when you add up a sequence of numbers starting from 1 and continuing consecutively, such as 1, 2, 3, all the way up to a certain number. For example, in the exercise, the task is to find the sum of all numbers from 1 to 100. This sequence is a special type of series known as an arithmetic series. To understand how to find such sums efficiently, we use specific mathematical formulas. This avoids adding individual numbers one by one, which can be tedious for large numbers. Here, we rely on the arithmetic series formula to easily calculate the sum.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant to the previous term. This constant is referred to as the common difference. - The sequence from 1 to 100, which increases by a common difference of 1, is a classic example of an arithmetic sequence.- In this context, the first term (\(a\),) is 1 and the last term (\(l\) ) is 100. - The number of terms (\(n\) ) is the count of numbers from 1 to 100, which is 100.Understanding these components helps to apply the formula for the sum effectively. Every arithmetic sequence can be summed by using established series formulas, making it a useful tool in mathematics.

Series Formula

The series formula is a mathematical formula used to find the sum of sequential terms in arithmetic series effectively. For an arithmetic sequence, this formula is\[ S = \frac{n}{2} [a + l] \]where

• \(S\) stands for the sum of the series.
• \(n\) represents the number of terms in the sequence.
• \(a\) is the first term of the series.
• \(l\) is the last term of the series.

By substituting these values into the formula, you can calculate the sum directly. As shown in the exercise example, substituting \(n = 100\), \(a = 1\), and \(l = 100\), yields \(S = 5050\). This formula is a cornerstone for solving problems involving arithmetic series and allows for swift computation without manual addition of each individual term.