Problem 117 - Algebra and Trigonometry

Question

Exercises \(116-118\) will help you prepare for the material covered in the next section. In Exercises \(116-117\) show that $$1+2+3+\cdots+n=\frac{n(n+1)}{2}$$ is true for the given value of \(n\) $$ n=5: \text { Show that } 1+2+3+4+5=\frac{5(5+1)}{2} $$

Step-by-Step Solution

Answer

Both sides of the equation provide the same result. Therefore, the formula \(1+2+3+...+n=\frac{n(n+1)}{2}\) holds for \(n=5\).

Step 1: Understanding the Formula

The given formula \(1+2+3+...+n=\frac{n(n+1)}{2}\) is used to find the sum of the first n natural numbers. For example, if n equals to 3, then the formula would represent the sum of the first three numbers 1, 2 and 3.

Step 2: Substituting the Value of n

The next step is to substitute the given value of \(n=5\) into the equation. This gives us \(1+2+3+4+5=\frac{5(5+1)}{2}\).

Step 3: Verifying the Equation

We simplify both sides individually. The left side of the equation is the sum of the first 5 natural numbers. This sums up to 15. The right side of the equation simplifies to \(\frac{5*6}{2}=15\). If both sides of the equation are equal, this confirms that the formula holds for n=5.

Key Concepts

Natural Number Summation

Understanding the summation of natural numbers is a foundational concept in algebra and provides a basis for more complex equations. Summation refers to the process of adding a sequence of numbers together. For natural numbers, which are the positive integers starting from 1, there's a specific pattern that we encounter.

Summing Consecutive Natural Numbers

The sum of the first few natural numbers is relatively simple to calculate by hand. For instance, the sum of the numbers 1 to 5 is found by simply adding each number: 1 + 2 + 3 + 4 + 5. As we want to deal with larger numbers, manually adding each consecutive term would be very time-consuming. Fortunately, there's a formula to find the sum quickly without listing all numbers: \(\sum_{i=1}^{n}i = \frac{n(n+1)}{2} \).

This equation means that the sum of the first n natural numbers is equal to half of the product of the number n and the next consecutive number (n+1). To use this formula, it's crucial to understand how to substitute values, as it can be applied to verify the sum for any specific value of n, just like the exercise demonstrates for n=5. Summation can be visualized by pairing numbers from the beginning and end of the range which also helps to rationalize why the formula works.

Mathematical Induction

Mathematical induction is a powerful technique used to prove the truth of an infinite number of cases in mathematics, such as formulas or theorems. Think of it as climbing an infinite ladder; if you can step onto the first rung (base case), and you can demonstrate that for every step you take, the next step can be reached (inductive step), then you can climb the ladder indefinitely.

Base Case and Inductive Step

The first part of the inductive proof is to verify that the statement is true for the initial case, often when n=1. This is called the base case. Once the base case is proven, the inductive step is to assume that the statement is true for some integer k, then prove it's also true for the next integer (k+1).

In the textbook exercise, if we extend the concept beyond n=5, we would utilize mathematical induction to prove that the sum formula holds true for all natural numbers n. This method assures that the formula does not only work for isolated examples but indeed for every natural number, which provides the necessary logical certainty to use it confidently in algebra.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term after the first is obtained by adding a constant difference. The series which sums natural numbers, as in the exercise example, is a particularly simple form of an arithmetic series.

Characteristics of an Arithmetic Series

The sequence of natural numbers (1, 2, 3, 4, 5, ...) has a common difference of 1. To find the sum of an arithmetic series, you could, in theory, add all the terms together, but this becomes impractical as the number of terms increases. That's where the summarized formula \(S_n = \frac{n}{2}(a_1 + a_n)\) comes in, where \(S_n\) represents the sum of the series, \(a_1\) is the first term, \(a_n\) the last term, and n is the number of terms.

The formula \(\frac{n(n+1)}{2}\) for the sum of the first n natural numbers is a specialized case of this arithmetic series formula, where the first term \(a_1\) is 1, and the last term \(a_n\) is n itself. Recognizing the connection between specific cases and the general patterns of arithmetic series is crucial for grasping broader algebraic concepts and solving complex problems effectively.