Problem 57 - Thinking Mathematically

Question

In Exercises 57-60, identify the reasoning process, induction or deduction, in each example. Explain your answer. It can be shown that $$ 1+2+3+\cdots+n=\frac{n(n+1)}{2} $$ I can use this formula to conclude that the sum of the first one hundred counting numbers, \(1+2+3+\cdots+100\), is $$ \frac{100(100+1)}{2}=\frac{100(101)}{2}=50(101) \text {, or } 5050 \text {. } $$

Step-by-Step Solution

Answer

The reasoning process used in this exercise is deductive reasoning because it starts with a general rule (the formula for the sum of a series) and applies it to a specific case (calculating the sum of the first 100 natural numbers).

Step 1: Examining the formulas

First, observe that the general formula \(1+2+3+...+n=\frac{n(n+1)}{2}\) is given, then the specific case is derived \(1+2+3+...+100=\frac{100(100+1)}{2} =5050\). So, it starts from a general formula and moves to a specific instance.

Step 2: Identify the reasoning process and justify

Since the reasoning here starts from a general principle or argument (the series sum formula) and is applied to a specific instance (the sum of the first 100 natural numbers), this is characteristic of deductive reasoning. In deductive reasoning, if the general principle is correct, then the derived conclusion must also be correct.

Key Concepts

Deductive Reasoning

When you start with a general rule and apply it to a specific case, you are using deductive reasoning. This form of reasoning ensures that if the starting point - called the premise - is true, then the conclusion is also guaranteed to be true. In our exercise, we begin with a known general formula for the sum of a series:

• The series sum formula: \(1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\)
• Apply to specific case \(n = 100\): \(1 + 2 + 3 + \cdots + 100 = \frac{100(100+1)}{2}\)

Here, we start from a broad mathematical truth and derive a particular instance, proving our conclusion is correct if our initial formula is correct. Deductive reasoning is powerful because it moves from the general to the specific.

Inductive Reasoning

Inductive reasoning takes a different route compared to its deductive counterpart. It involves observing specific examples and identifying a pattern that helps formulate a general rule. While it provides a way to make generalizations, conclusions from inductive reasoning have a degree of uncertainty and are not guaranteed to be true. For example:

• Imagine summing smaller series like \(1 + 2 + 3 = 6\) and \(1 + 2 + 3 + 4 = 10\).
• Notice that the results could fit the equation \(\frac{n(n+1)}{2}\).

By observing specific patterns, you might hypothesize the series sum formula itself through induction. However, unlike deduction, these conclusions need further proof or testing to be certain.

Series Sum Formula

The series sum formula provides a succinct way to calculate the sum of consecutive natural numbers up to \(n\). Mathematically stated, it is:\[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\]This formula was derived from the observation of patterns in numbers and validated through mathematical deduction. It is extremely useful as it saves time by providing a direct route to the sum instead of adding each number individually. For instance, using it means we do not have to manually add numbers from 1 to 100 to find out the sum is 5050. Instead, we efficiently plug in 100 for \(n\):\[\frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 5050\] Utilizing formulas like these exemplifies how mathematics provides efficient, reliable tools for problem-solving.

Mathematical Deduction

Mathematical deduction involves deriving conclusions strictly from general principles through a logical step-by-step process. It heavily relies on previously established truths or axioms to ensure the correctness of the conclusion. Unlike induction, which might suggest the formula based on observed examples, deduction insists that it must be provably true if derived logically. When you use the series sum formula on a specific number like in our example (\(n = 100\)), each step connects clearly to the established general principle. Deductively, if the formula is correct, then every specific instance derived from it will be correct as well. This gives mathematics its reliability, as deduction ensures certainty in results whenever its methods are correctly applied.