Problem 84 - Algebra and Trigonometry

Question

Show that the sum of the first \(n\) positive odd integers, $$ 1+3+5+\dots+(2 n-1) $$ is n2.

Step-by-Step Solution

Answer

The exercise's assertion, that the sum of the first \(n\) positive odd integers is \(n^2\), is proven through mathematical induction.

Step 1 - Base Case

We need to verify the given formula for \(n = 1\). If \(n = 1\), then the left side of the formula is \(1\), and the right is \(1^2 = 1\). So the given formula is true for \(n = 1\).

Step 2 - Inductive hypothesis

Now, we assume that the given formula is true for \(n = k\). That is, we assume that \(1 + 3 + 5 + ... + (2k - 1) = k^2\).

Step 3 - Inductive step

We need to prove that the formula will be correct for \(n = k+1\), assuming our inductive hypothesis is true. Adding the next term \((2(k+1)−1)\) to both sides, will give us \(k^2 + 2(k+1) - 1\) on the right side. Simplify this to \((k+1)^2\). Hence the sum of the first \(k+1\) odd positive integers is equal to \((k+1)^2\), proving our formula. This concludes the inductive step.

Key Concepts

Inductive Hypothesis

In mathematical induction, the inductive hypothesis is a critical step in the proof process. It involves assuming that a given statement or formula is true for a specific natural number, usually denoted as \(n = k\).
This assumption is not made to be questioned; instead, it acts as a stepping stone to prove the formula's validity for the next integer, \(n = k+1\).

• The inductive hypothesis simplifies problems by assuming our formula holds for a certain case.
• For example, in our exercise, we assume the formula \(1 + 3 + 5 + \, ... \, + (2k - 1) = k^2\) holds true.
• This assumption lets us focus on proving the formula holds for subsequent numbers.

By basing on this hypothesis, we extend the truth of the proposition step by step, which is a powerful aspect of mathematical proof.

Base Case

Every proof by induction begins with the base case, which serves as the foundation of the entire process. The base case involves verifying the proposition for the initial value of \(n\), usually \(n = 1\).
Establishing the base case is crucial because it confirms the starting point of the induction process. Without a valid base case, the whole induction may falter.

• In the exercise, the base case checks if \(1 = 1^2\).
• The left side, which is the first odd integer, is verified to be equal to the square of \(n\).
• A correct base case means the formula holds true at least for \(n = 1\).

This setup is vital as it initiates the chain of logic that extends through all natural numbers.

Inductive Step

The inductive step is where mathematical induction proves its power by extending the truth of the formula for one number to the next.
Assuming the formula holds true for \(n = k\) (by the inductive hypothesis), the goal here is to show it must also be true for \(n = k+1\).

• In our exercise, we add the next odd integer, \((2(k+1)-1)\), to the assumed true sum \(k^2\).
• This transforms the sum to \(k^2 + (2k + 1) = (k+1)^2\).
• The transformation confirms the formula is valid for \(n = k+1\).

The success of the inductive step is essential as it ensures the validity of the formula across all natural numbers following the base case.

Sum of Odd Integers

The sum of the first \(n\) positive odd integers is a well-known exploration in mathematics.
It showcases a beautiful, straightforward result: \(1 + 3 + 5 + \dots + (2n-1) = n^2\).

• Understanding begins with recognizing the pattern in odd numbers: they increment by 2.
• The formula \(n^2\) signifies how accumulation of these odds forms a perfect square.
• This sum can represent geometric figures like squares, fostering a visual understanding.

This concept not only highlights numerical patterns but also enriches comprehension through abstract mathematical relations.