Question

Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters \(4-7\). If \(n \in \mathbb{N},\) then \(2^{0}+2^{1}+2^{2}+2^{3}+2^{4}+\cdots+2^{n}=2^{n+1}-1\)

Short answer

The given formula is true for all n in the set of natural numbers, proven using the principle of mathematical induction.

Step-by-step solution

Problem Statement Verification

Using substitution, it can be confirmed that the base case (n=0) is true. The left side becomes \(2^{0}\) which equals 1 and the right side reduces to \(2^{1}-1\) which also equals 1, thus the base case holds.

Induction Hypothesis

Assume that the given formula is true for some arbitrary non-negative integer, k. That means \(2^{0}+2^{1}+2^{2}+2^{3}+\cdots+2^{k}=2^{k+1}-1\)

Inductive Step

The goal is to prove that the formula holds true for k+1. Adding \(2^{k+1}\) to both sides of the equation results in \(2^{0}+2^{1}+2^{2}+2^{3}+\cdots+2^{k}+2^{k+1}=2^{k+1}-1+2^{k+1}\). The right side simplifies to \(2*(2^{k+1})-1=2^{k+2}-1\). This is the formula for n=k+1, which shows that if the formula is true for n=k, it is also true for n=k+1.

Key concepts

Proof Techniques

Understanding proof techniques is fundamental to appreciating and performing mathematical validation. One common approach to proving theorems or statements, particularly those involving natural numbers, is mathematical induction. This technique has two key steps.

First, the base case establishes the truth of the statement for an initial value, typically 0 or 1. Think of it as the foundation of a domino effect. In our problem, when we substitute n = 0, both sides of the equation balance, confirming the base case.

The second step is the inductive step. We assume that the statement is true for some arbitrary value k (induction hypothesis), and then prove it's also true for the next value, k + 1. By adding the term for k + 1 to both sides and verifying the equality continues to hold, we essentially knock over the next domino. This 'domino effect' implies the statement is true for all subsequent natural numbers.

This procedure encapsulates the very essence of mathematical induction—one of the powerful proof techniques used in mathematics to establish a proposition's truth beyond any reasonable doubt.

Sequences and Series

In mathematics, a sequence is an ordered list of numbers and a series is the sum of the elements of a sequence. Sequences can be finite or infinite, depending on the number of terms it contains.

In our exercise, we are dealing with the finite sequence of the form \(2^{0}, 2^{1}, 2^{2}, ..., 2^{n}\). The series for this sequence is the sum of all its terms \(2^{0} + 2^{1} + 2^{2} + ... + 2^{n}\). This particular series is a sum of powers of 2, which grows exponentially with each additional term. Sequences and series are pervasive in mathematics and form the backbone of various fields, including calculus and financial mathematics.

The proof that we're discussing shows how the series sum formula can be derived and verified using mathematical induction, demonstrating the interconnectedness of various concepts such as sequences, series, and proof techniques in mathematical analysis.

Exponential Functions

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It is easily identifiable by its form \(b^{x}\), where b is the base and x is the exponent. In our example, the base is 2, and the exponent is n.

Exponential functions are recognized for their rapid growth or decay properties. They are fundamental to various disciplines, including biology, economics, and physics, making them a topic of great importance and interest in mathematics.

The geometric progression in the presented problem is a real-world example of how exponential growth behaves and how it can be represented through the sum of an exponential series. The formula \(2^{n+1} - 1\) on the right-hand side of the equation exemplifies the aggregate growth exhibited by the series of powers of 2, encapsulating the nature of exponential functions. Understanding exponential functions not only aids in solving series problems but also in grasping complex phenomena across different scientific fields.