Problem 17 - Algebra and Trigonometry

Question

Use mathematical induction to prove that each statement is true for every positive integer n. \(1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1\)

Step-by-Step Solution

Answer

The statement \(1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1\) holds true for every positive integer \(n\), as we have successfully proven using mathematical induction.

Step 1: Base Case

The base case tests whether the statement holds for \(n=1\). When \(n=1\), left hand side equals 1 and the right hand side equals \(2^{1}-1=1\). Thus, the statement is true for \(n=1\).

Step 2: Induction Hypothesis

Assume that the statement is true for some positive integer \(k\), which means \(1+2+2^{2}+\cdots+2^{k-1} = 2^{k}-1\).

Step 3: Induction Step (Proof)

We need to show that the statement holds for \(n=k+1\), adding the \(k+1^{th}\) term to both sides of the equation, we get \(1+2+2^{2}+\cdots+2^{k-1}+2^{k} = 2^{k}-1+2^{k}\). This simplifies to \(2^{k+1}-1\), which is the right-hand side of the equation for \(n=k+1\). Therefore, the statement holds true for all positive integers \(n\).

Key Concepts

Proof by Induction

Mathematical induction is a powerful proof technique used to establish the truth of a statement for all positive integers. It involves two main steps: the base case and the induction step. The base case verifies the statement for the smallest value, usually with the integer 1. Once confirmed to be true, the induction hypothesis assumes that the statement is true for a certain integer, say \( k \). Next, the induction step shows that if the statement holds true for \( k \), it must also be true for \( k+1 \).

This approach works effectively for proving formulas related to sequences, series, and other mathematical expressions that rely on natural number patterns. The beauty of proof by induction lies in its ability to bridge a specific case (base case) to a universal application across infinite instances (all positive integers).

Mathematical Proof Techniques

Mathematical proofs are structured arguments that use logic to demonstrate the truth of a mathematical statement. There are various proof techniques including:

• Direct proof: Demonstrates the truth of a statement by straightforward logical deduction.
• Contradiction: Assumes the opposite of what you want to prove and shows it leads to a contradiction.
• Induction: As discussed, it is ideal for statements involving natural numbers.
• Contrapositive: Proves a statement by proving its contrapositive is true.

Each technique has its own applications and strengths, with induction particularly suited for statements involving sequences and series. A solid understanding of these methods enriches problem-solving skills and mathematical comprehension.

Sequences and Series

Sequences are lists of numbers following a certain rule, while a series is the summation of the terms in a sequence. The given problem is an example where we sum the terms of a sequence formed by powers of 2: \(1, 2, 2^2, \ldots, 2^{n-1}\).

In our formula for this series, \(1 + 2 + 2^2 + \cdots + 2^{n-1}\), the expression simplifies to \(2^n - 1\). Recognizing this pattern is key in applying mathematical induction to prove the formula for any positive integer \( n \). Understanding series helps in making predictions about larger sums without needing to add each number straightforwardly.

Positive Integers

Positive integers are the set of all natural numbers that begin from 1 and extend infinitely: \(1, 2, 3, \ldots\). They are critical in mathematics as they form the foundation of counting and ordering. Many mathematical statements are formulated considering positive integers because they provide a straightforward context for computation and analysis.

In proofs by induction, we're typically dealing with positive integers because they lend themselves well to logical, sequential arguments — like stepping stones from one true statement to another. They ensure that patterns hold indefinitely, making them perfect candidates for proving propositions using induction.