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The base case in mathematical induction is the starting point that proves a property for the smallest possible value of the positive integer \( n \). It's crucial to establish this anchor because it's like laying the foundation of a building. If this initial case does not hold true, the validity of subsequent steps is questionable.

In this exercise, we look at \( n = 1 \). Substituting \( n = 1 \) into the given formula, the left-hand side (LHS) becomes simply 1. On the right-hand side (RHS), we calculate \( 2^1 - 1 = 1 \). Since both LHS and RHS are equal, the base case holds true, reinforcing the validity of our formula for \( n = 1 \). This step is often straightforward, but it's a critical confirmation that sets the stage for the rest of the proof.

The inductive step is where mathematical induction gets powerful. This step involves two parts: assuming the formula is true for a positive integer \( k \), and then using this assumption to prove it's true for the next integer, \( k+1 \).

Let's first understand what the induction step entails here:

• Assumption (Inductive Hypothesis): We accept that our formula holds for \( n = k \), meaning \( 1 + 2 + 2^2 + \cdots + 2^{k-1} = 2^k - 1 \).
• Proof for \( k+1 \): We now aim to verify that the formula is true when \( n = k+1 \). If the formula holds for \( n = k \), then adding \( 2^k \) to both sides, we have \( 1 + 2 + 2^2 + \cdots + 2^{k-1} + 2^k = 2^k - 1 + 2^k = 2 \times 2^k - 1 \), which simplifies to \( 2^{k+1} - 1 \).

This step successfully connects \( n = k \) to \( n = k+1 \), showcasing the cascading effect where if one domino falls, the majority of them will follow. This linkage is the heart of the inductive method.

The inductive hypothesis is essentially the assumption that the property or formula we aim to prove holds true for a specific, but arbitrary, positive integer \( k \). It's vital as it becomes the bridge to establish truth for the next integer, \( k+1 \).

In this exercise, our inductive hypothesis is that the formula \[ 1 + 2 + 2^2 + \cdots + 2^{k-1} = 2^k - 1 \] is assumed to be true. This hypothesis is not something to be proven at this stage; instead, its role is hypothetical and helps in extending the proof to the next integer value. By proving that if the hypothesis works for \( n = k \), it also works for \( n = k+1 \), the assumption justifies itself in a cyclical fashion. This logical sequence ensures that the formula is valid for all positive integers starting from our base case.