Problem 16 - College Algebra

Question

In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) $$ 2+7+12+\dots+(5 n-3)=\frac{n(5 n-1)}{2} $$

Step-by-Step Solution

Answer

Using mathematical induction, it has been proven that the statement \(2+7+12+\dots+(5 n-3)=\frac{n(5 n-1)}{2}\) holds true for every positive integer \(n\).

Step 1: Base Case

Let's verify the base case for \(n = 1\). The left side of the equation becomes \(5(1) - 3 = 2\) and the right side becomes \(\frac{1(5(1) - 1)}{2} = 2\). As this holds true, the base case is proven.

Step 2: Inductive Hypothesis

Assume the hypothesis \(2 + 7 + 12 + \dots + (5k - 3) = \frac{k(5k - 1)}{2}\) is true for some positive integer \(k\). This is our inductive hypothesis.

Step 3: Inductive Step

Using this assumption, we need to prove the statement for \(n = k + 1\). Adding the \((k+1)\) term to both sides of the hypothesis yields: \(2 + 7 + 12 + ...+ (5k - 3) + (5(k + 1) - 3) = \frac{k(5k - 1)}{2} + (5(k + 1) - 3)\). Simplifying the right side gives: \[\frac{k(k+1)(5k+4)}{2}\]. This simplifies to the equation expected in the prompt, \[\frac{(k+1)(5(k+1) - 1)}{2}\], therefore the inductive step is proven and we have used mathematical induction to prove the proposition is true for every positive integer \(n\).

Key Concepts

Base Case

In the method of mathematical induction, the base case is the starting point of the proof. It verifies that the given proposition holds true for the first value in the sequence of positive integers, typically starting with 1.

In our specific example, the expression to be proven is \(2+7+12+\dots+(5n-3)=\frac{n(5n-1)}{2}\). First, we test this expression for \(n=1\). Replacing \(n\) with 1, the left-hand side evaluates to \(5 \times 1 - 3 = 2\). Similarly, the right-hand side becomes \(\frac{1(5 \times 1 - 1)}{2} = 2\). Since both sides equal, the base case is verified as true.

Establishing a true base case creates a foundation for the subsequent steps in the induction process.

Inductive Hypothesis

The inductive hypothesis is an assumption made during the process of mathematical induction. After establishing the base case, you assume that the statement is true for some arbitrary positive integer, typically denoted by \(k\).

In our situation, the inductive hypothesis assumes that the statement \(2 + 7 + 12 + \dots + (5k - 3) = \frac{k(5k - 1)}{2}\) is true. This assumption is not meant to be proven in this step; rather, it serves as a stepping stone to prove the next step.

Assumptions made in the inductive hypothesis are crucial for reaching the next phases of the proof.

Inductive Step

In the inductive step, you leverage the assumption from the inductive hypothesis to prove that the statement is true for the next integer, often written as \(k + 1\). If this is shown to be true, then the statement holds for all positive integers in the sequence.

For our example, we need to demonstrate the truth of \(2 + 7 + 12 + \dots + (5(k+1) - 3)\). Starting with the hypothesis, we add the term corresponding to \(k+1\), which is \(5(k+1) - 3\). This yields:

• \(2 + 7 + 12 + \dots + (5k - 3) + (5(k + 1) - 3)\)
• \(= \frac{k(5k - 1)}{2} + (5(k + 1) - 3)\)

Simplifying, this provides \[\frac{k(k+1)(5k+4)}{2}\], matching the desired formula \(\frac{(k+1)(5(k+1) - 1)}{2}\). Thus, the inductive step successfully demonstrates the truth of the statement for \(n=k+1\).

Positive Integer

Mathematical induction is specifically designed to work over positive integers, which are the numbers \(1, 2, 3,\) and so forth. Each proof using induction involves demonstrating that a statement is true over these numbers, starting from the base case of \(n=1\).

A positive integer is crucial in the context of induction because these are natural numbers where sequencing and progression make logical sense.

In our example, by proving the base case for \(n=1\) and using the inductive step to confirm it for \(n=k+1\), the statement is verified to be true for all positive integers. This progression reflects the very structure of mathematical induction as a proof method that relies on the properties of numbers in a continuous set.