Problem 20 - College Algebra

Question

In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}} $$

Step-by-Step Solution

Answer

The given equation \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}\) holds true for all positive integers \(n\) as indicated by mathematical induction. The results of the Induction hypothesis and the Induction step prove this.

Step 1: Initial Hypothesis

Start with the base case, where \(n=1\). The left-hand side (LHS) of the given equation is \(\frac{1}{2}\), and the right-hand side (RHS) is \(1-\frac{1}{2^1} = \frac{1}{2}\). So, the statement holds true for \(n=1\)

Step 2: Induction Step I

Next, assume that the equation holds true for \(n=i\), where \(i\) is a generic integer. This gives us the following equation: \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^{i}} = 1 - \frac{1}{2^{i}}\). This is called the Inductive hypothesis.

Step 3: Induction Step II

Then demonstrate that if the equation holds true for \(n=i\), it must hold for \(n=i+1\). Add \(\frac{1}{2^{i+1}}\) to both sides of the equation from step II: LHS = \(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^{i}} + \frac{1}{2^{i+1}}\) and RHS = \(1 - \frac{1}{2^{i}} + \frac{1}{2^{i+1}}\). Rewrite the RHS as \(1 - \frac{1}{2^{i+1}}\), so we get the same expression for \(n=i+1\) showing, the equation holds true for all \(n\).

Key Concepts

Proof by Induction

Mathematical induction is a powerful technique to prove the validity of an assertion for all positive integers. It is akin to a domino effect; if you knock over the first one and prove that any one knocking over will knock over the next, you can ensure all subsequent ones will fall. Think of induction in two critical steps: the base case and the inductive step.

In the base case, you verify the statement for the initial value, usually when \( n = 1 \). This serves as the stepping stone for the process. The inductive step involves assuming the statement is true for a generic integer \( i \) and then proving it must also be true for \( i + 1 \). The beauty of this step is it creates a logical chain that extends ad infinitum – proving the statement for all positive integers.

Sequences and Series

Sequences and series are foundational concepts in algebra that provide order and summation to sets of numbers. A sequence is a list of numbers in a specific order, often following a rule or pattern. Meanwhile, a series is the sum of the elements of a sequence.

In our exercise, the series is the sum of terms in the form \( \frac{1}{2^n} \), where \( n \) is a positive integer. As \( n \) increases, the terms get progressively smaller, revealing a pattern characteristic of a geometrical sequence. The infinite sum (or the limit of the series as \( n \) approaches infinity) can often be expressed in a simple formula, which, in educational contexts, aids greatly in the understanding of the behavior of sequences over time.

Algebraic Proofs

Algebraic proofs are logical arguments that utilize algebraic techniques to demonstrate the truth of a statement. Crucially, algebraic proofs are not mere calculations; they are reasoned narratives that use equations and inequalities as their language.

Incorporating a known formula, manipulation of terms, or understanding of algebraic identities forms the basis of these proofs. Even complex mathematical theorems often boil down to a series of algebraic steps that, when taken incrementally, lead to a proof's conclusion. In our provided solution, we utilize algebraic manipulation to transform and compare the expressions for steps II and III, thus confirming the truth of the inductive hypothesis.

Positive Integers

Positive integers are the set of all whole numbers greater than zero. They are the basic building blocks for many areas of mathematics and are especially important in proofs by induction. This importance arises because mathematical induction is designed to prove statements for these counting numbers, starting from 1 and moving upward without end.

The beauty of focusing on positive integers is that each has a distinct successor, which is perfect for the 'step-by-step' nature of induction. When dealing with proofs involving these numbers, we're often looking at properties that are inherited as we move from one integer to the next - as seen in the exercise regarding the sum of a series.