Problem 17 - College Algebra

Question

In Exercises 11-24, 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

By using the principle of mathematical induction, it is found that the given formula is valid for all positive integers, exactly as required.

Step 1: Base Case

Verify that the statement holds true for \(n = 1\). Substituting \(n = 1\) into both sides of the equation gives \(2^{1} - 1 = 1\) on the right-hand side which is true as \(1 = 1\), thus satisfying the base case.

Step 2: Inductive Step - Assume true for \(n = k\)

Assume that the statement is true for \(n = k\), i.e., \(1 + 2 + 2^{2} + ... + 2^{k - 1} = 2^{k} - 1\). This is the Inductive Hypothesis.

Step 3: Inductive Step - Prove true for \(n = k + 1\)

We seek to show the statement is valid for \(n = k + 1.\) Starting with the left-hand side of the expression for \(n = k + 1,\) we get \(1 + 2 + 2^{2} + ... + 2^{k - 1} + 2^{k}.\) Replace the sum \(1 + 2 + 2^{2} + ... + 2^{k - 1}\) with \(2^{k} - 1\) from the inductive hypothesis. This results in \((2^{k} - 1) + 2^{k} = 2 \cdot 2^{k} - 1 = 2^{k+1} - 1,\) which is the required form for \(n = k + 1.\) Therefore, the assumption that the statement holds for \(n = k\) has allowed us to prove that it also holds for \(n = k + 1\).

Step 4: Conclusion

Since the formula holds true for the base case \(n = 1\) and if it holds true for \(n = k\) then it holds true for \(n = k + 1,\) by the principle of mathematical induction, the given formula holds for all positive integers.

Key Concepts

Positive Integers

Positive integers, commonly referred to as natural numbers or counting numbers, are the numbers starting from 1 and increasing by 1 each time (e.g., 1, 2, 3, ...). Understanding positive integers is fundamental when exploring mathematical concepts such as series and sequences, which often involve summing terms indexed by these numbers. For instance, in the exercise provided, the sum involves terms that are powers of 2, with the exponents being consecutive positive integers less than or equal to a given number, denoted here as 'n'.

Moreover, in the context of sequences (a type of series), positive integers are used to describe positions within the series. Position 1 corresponds to the first term, position 2 to the second, and so on. The position is significant as each term is connected to its position by a distinct mathematical rule or pattern; a relationship is clearly illustrated in our textbook exercise.

Series and Sequences

In mathematics, a sequence is an ordered list of numbers that follow a particular pattern, while a series is the sum of the terms of a sequence. Sequences are omnipresent in mathematics and can be either finite or infinite, depending on whether they have a last term or not. Conversely, series can extend infinitely or have a finite sum, often depending on the nature of the sequence it's derived from.

The series presented in the textbook problem is a finite geometric series because each term after the first is obtained by multiplying the previous term by a constant ratio, here the constant being 2. The problem challenges students to prove that the sum of the series up to any positive integer 'n' follows a particular formula. Understanding how these terms accumulate and conform to specific rules provides a better grasp of not just this series, but series and sequences in general, as such structures form the bedrock for numerous mathematical theories and applications.

Proof Technique

Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of propositions. With its origins dating back centuries, it remains a pillar of mathematical logic and proof theory. This approach follows a simple yet effective two-step process: the 'base case' and the 'inductive step'.

Firstly, in the base case, one shows that the statement holds true for the first positive integer (commonly 1). This step is crucial as it acts as the foundation for the argument. Next, in the inductive step, one assumes the statement is true for some positive integer 'k' (the inductive hypothesis) and then uses this assumption to prove that the statement is also true for 'k+1'. The beauty of induction lies in its simplicity and universality - if these two steps can be completed, the statement must hold for all positive integers, as each integer will be linked by the same domino effect that is initiated from the base case.

Applying mathematical induction to our textbook exercise, we can confidently extend the truth of the formula from a single instance to all positive integer values of 'n', creating a bridge from the known to the unknown, one integer at a time.