Problem 26 - College Algebra

Question

In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) 2 is a factor of \(n^{2}+3 n\)

Step-by-Step Solution

Answer

Using the principle of mathematical induction, it has been proved that 2 is a factor of \(n^{2}+3 n\) for every positive integer \(n\).

Step 1: Base Case (n=1)

Substitute \(n=1\) into the equation. This gives \(1^{2}+3*1 = 4\). Since 4 is divisible by 2, the statement is true for \(n=1\).

Step 2: Inductive Step

Assume that the statement is true for some integer \(k \geq 1\), meaning that 2 divides \(k^{2}+3k\), or stated otherwise, \(k^{2}+3k = 2m\) for some integer m. Now we have to prove the statement for \(k+1\). Substitute \(n=k+1\) into the equation to get \((k+1)^{2}+3(k+1)\). This simplifies to \(k^{2}+2k+1+3k+3 = k^{2}+5k+4\). By the inductive hypothesis \(k^{2}+3k=2m\), so \(k^{2}+5k+4 = 2(m+k+2)\). As \(m\), \(k\) and 2 are integers, \(m+k+2\) is an integer. Thus, the term \(k^{2}+5k+4\) is divisible by 2, which proves the statement for \(n=k+1\). This completes the induction step.

Step 3: Conclusion

By the Principle of Mathematical Induction, we've shown the statement is true for \(n=1\) and assuming it's true for \(n=k\), then it must also be true for \(n=k+1\). Hence, the statement that 2 is a factor of \(n^{2}+3 n\) is true for all positive integers \(n\).

Key Concepts

Factor of Polynomial

Understanding what a factor of a polynomial is can be crucial when you're trying to prove a statement like '2 is a factor of n^2 + 3n'. A factor of a polynomial is an expression that can be multiplied by another expression to get the original polynomial. In simpler terms, if you can find a number or another polynomial that divides into the original polynomial without leaving a remainder, then you've found a factor. In the problem at hand, showing that 2 is a factor involves demonstrating that when any positive integer is plugged into n^2 + 3n, the result is divisible by 2.

To give you a clearer example, if we plug in n = 1, we get 1^2 + 3(1) = 4, which is clearly divisible by 2, indicating that 2 is indeed a factor of the polynomial for n = 1. It's much like breaking a chocolate bar into two equal parts - the bar represents the polynomial and each half represents a factor. Remember, the factors are the 'building blocks' that multiply together to create the original polynomial.

Proof by Induction

Proof by induction is a powerful mathematical tool often used to prove the validity of statements for all natural numbers. It works a bit like dominoes; you knock the first one over (the base case) and show that if one domino falls (the inductive step for some k), the next one will too (the step for k+1). Thus, it creates an endless chain of truth.

For the exercise, the first step is to demonstrate that the statement is true when n = 1, known as the base case. Once this is established, we approach the inductive step. We assume that if the statement holds for a certain integer k, then it also holds for k+1. If we manage to prove that this step works, we've cemented the domino effect, confirming the statement for all positive integers thereafter. This logical progression from one case to the next is what gives mathematical induction its convincing power - and its beauty.

Divisibility

Divisibility is a pretty straightforward concept: one number is divisible by another if you divide them and there's no remainder. Think of it like sharing cookies evenly among friends; if you have exactly enough cookies so that no one is left cookie-less and there are none left over, the number of cookies is divisible by the number of friends.

In terms of our exercise, to show that n^2 + 3n is divisible by 2, we're essentially proving that this expression, for any positive integer n, can be split into two equal parts - every time, no leftovers. For instance, when n=k+1, substituting this into the expression gives us an even result, meaning it can be divided evenly by 2. This divisibility is a key part of mathematical induction and helps confirm the truth of the statement we're working with for every positive integer.