Problem 8 - Algebra and Trigonometry

Question

Use mathematical induction to prove that the formula is true for all natural numbers \(n\). $$ 1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2 n+7)}{6} $$

Step-by-Step Solution

Answer

The formula is true for all natural numbers \( n \).

Step 1: Base Case

To begin with mathematical induction, we first verify the base case. We'll test the formula for the smallest natural number, usually \( n = 1 \).Substituting \( n = 1 \) into the left-hand side of the equation gives:\[1 imes 3 = 3\]And, for the right-hand side:\[\frac{1(1+1)(2\times1 + 7)}{6} = \frac{1 imes 2 imes 9}{6} = 3\]The base case holds true as both sides equal 3.

Step 2: Inductive Hypothesis

Assume that the statement is true for some natural number \( k \). That is, assume:\[1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + k(k+2) = \frac{k(k+1)(2k+7)}{6}\]This assumption is known as the inductive hypothesis.

Step 3: Inductive Step

Now, we need to prove that if the statement holds for \( n = k \), then it must also hold for \( n = k + 1 \). That is, we need to prove:\[1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + k(k+2) + (k+1)(k+3) = \frac{(k+1)((k+1)+1)(2(k+1)+7)}{6}\]Start by adding \( (k+1)(k+3) \) to both sides of the inductive hypothesis:\[\frac{k(k+1)(2k+7)}{6} + (k+1)(k+3)\]

Step 4: Simplify the Inductive Step

Combine the terms carefully:Factor out \( k+1 \) from the terms:\[(k+1) \left( \frac{k(2k+7)}{6} + (k+3) \right)\]Find a common denominator and simplify:\[(k+1) \left( \frac{2k^2 + 7k + 6k + 18}{6} \right)\]Simplify the numerator:\[(k+1) \left( \frac{2k^2 + 13k + 18}{6} \right)\]

Step 5: Conclude the Induction

Finally, simplify to match the right-hand side of the formula for \( n = k+1 \):The expression simplifies to:\[\frac{(k+1)(k+2)((2k+2)+7)}{6} = \frac{(k+1)(k+2)(2k+9)}{6}\]This is exactly what we needed, confirming the formula holds for \( n = k+1 \). Therefore, by mathematical induction, the formula is true for all natural numbers \( n \).

Step 6: Conclusion

Since the base case and the inductive step both hold true, by the principle of mathematical induction, the formula \( 1 \cdot 3+2 \cdot 4+3 \cdot 5+\cdots+n(n+2)=\frac{n(n+1)(2n+7)}{6} \) is valid for all natural numbers \( n \).

Key Concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1. They are the numbers we naturally use to count objects, like 1, 2, 3, and so forth.
Natural numbers are infinite, and they do not include zero, negative numbers, or fractions.

• The first natural number is 1.
• Each number has a successor, which is the number that comes after it in the sequence.
• They are used in many areas of mathematics including arithmetic and number theory.

In the exercise, we're using natural numbers to apply a formula and prove its validity using mathematical induction.

Inductive Hypothesis

The inductive hypothesis is a critical assumption in mathematical induction.
It is the step where we assume that a statement is true for some arbitrary natural number, usually denoted by the variable \( k \).
Making this assumption allows us to bridge the gap between knowing the formula works for one number and proving it works for all.

• This assumption lets us build the foundation of our proof by establishing a known case.
• It acts as a hypothesis that aids in proving cases that come afterward, such as \( n = k + 1 \).
• In our exercise, we assume that the formula holds for \( n = k \).

By doing so, we're setting up the groundwork before testing the next natural number in our sequence.

Inductive Step

The inductive step is the process where we prove that if the formula is true for some number \( k \), then it also holds for the next number, \( k + 1 \).
This is the heart of mathematical induction, showing the domino effect from \( k \) to \( k + 1 \).

• Here, we take the assumed truth of \( n = k \) from the inductive hypothesis.
• Next, we add or adjust our calculations to test \( n = k + 1 \).
• In the example, we included additional terms for \( n = k + 1 \) and simplified them to ensure they align with the formula.

Ultimately, this step connects the base case to all subsequent cases by proving the chain-link connection from one number to the next.

Algebraic Formula

An algebraic formula represents a mathematical relationship using variables and constants.
In problems involving induction, these formulas express a pattern or sequence.
The formula given in the exercise is:\[1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + n(n+2) = \frac{n(n+1)(2n+7)}{6}\]

• It involves a sequence of products on the left side of the equation.
• The right side is written as a fraction, simplifying the sequence.

The goal of using induction with this formula is to prove its correctness for all natural numbers, showing that each term on the left corresponds precisely to the algebraic expression on the right when \ simplified for each natural number \( n \).
Understanding how these formulas work is crucial for successfully applying mathematical induction.