Let Sn represent the given statement, and use mathematical...

Let $S_{n}$ represent the given statement, and use mathematical induction to prove that $S_{n}$ is true for every positive integer $n .$ See Example $1 .$ Follow these steps. (a) Verify $S_{1}.$ (b) Write $S_{k}.$ (c) Write $S_{k+1}.$ (d) Assume that $S_{k}$ is true and use algebra to change $S_{k}$ to $S_{k+1}.$ (e) Write a conclusion based on Steps ( $a$ ) - ( $d$ ). $$\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Let $S_{n}$ represent the given statement, and use mathematical induction to prove that $S_{n}$ is true for every positive integer $n .$ See Example $1 .$ Follow these steps.
(a) Verify $S_{1}.$
(b) Write $S_{k}.$
(c) Write $S_{k+1}.$
(d) Assume that $S_{k}$ is true and use algebra to change $S_{k}$ to $S_{k+1}.$
(e) Write a conclusion based on Steps ( $a$ ) - ( $d$ ).
$$\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\dots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$$

Key Concepts

Mathematical Induction

Mathematical induction is a method of proof used to establish that a statement holds for all positive integers. It involves two major steps: first, verifying that the statement holds for an initial case (often n = 1), and second, proving that if the statement holds for an arbitrary integer (n = k), then it also holds for the next integer (n = k+1). This method is fundamental for proving propositions that are valid for an infinite sequence of cases.

Base Case Verification

The base case verification is the first step in a mathematical induction proof. It requires demonstrating that the given statement is true for the initial integer, commonly n = 1. This establishes the foundation of the induction argument by confirming that the starting point of the sequence satisfies the proposition.

Inductive Hypothesis

The inductive hypothesis is the assumption that the statement is true for an arbitrary positive integer n = k. This assumption is used as a provisional step to facilitate the proof of the statement for the next case (n = k+1). It is a crucial component that links the base case to the inductive step.

Inductive Step

The inductive step is the part of the proof where one shows that if the statement holds for an arbitrary integer n = k (as assumed in the inductive hypothesis), then it logically follows that the statement is also true for n = k+1. This step typically involves algebraic manipulation and logical reasoning to extend the validity of the statement from one integer to the next.

Algebraic Manipulation

Algebraic manipulation is a key technique used during the inductive step to transform the expression assumed for n = k (from the inductive hypothesis) into the expression needed for n = k+1. This involves algebraic techniques such as factoring, adding or subtracting terms, and simplifying expressions to demonstrate the transition from the kth case to the (k+1)th case.

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