Proof By Induction 8 Divides 7^(2n+1) + 1

Introduction to Proof by Induction

In the realm of mathematical proofs, induction stands as a powerful technique for establishing the truth of a statement across an infinite range of cases. It is particularly useful when dealing with statements that involve natural numbers. The principle of mathematical induction is akin to a domino effect: if you can show that the first domino falls (the base case) and that any falling domino will knock over the next one (the inductive step), then you can confidently assert that all the dominoes will fall. This article delves into how we can harness the power of induction to demonstrate that for all non-negative integers n, 8 is a factor of ${7^{2n+1} + 1}$.

The core idea behind mathematical induction can be broken down into three fundamental steps. First, the base case must be established. This involves showing that the statement holds true for the initial value, typically n = 0 or n = 1. It serves as the foundation upon which the rest of the proof is built. Secondly, the inductive hypothesis is assumed. Here, we assume that the statement is true for some arbitrary integer k. This assumption is crucial as it allows us to bridge the gap between one case and the next. Finally, the inductive step is performed. In this step, we must demonstrate that if the statement holds true for n = k, then it must also hold true for n = k + 1. This step essentially shows that the truth of the statement propagates from one value to the next. By successfully completing these three steps, we can confidently conclude that the statement is true for all natural numbers greater than or equal to the base case.

Mathematical induction is not merely a theoretical tool; it has practical applications across various domains of mathematics and computer science. For instance, it can be used to prove the correctness of algorithms, establish formulas for sequences and series, and even demonstrate fundamental properties of numbers. Its versatility and rigor make it an indispensable technique in any mathematician's toolkit. In the subsequent sections, we will apply this method to the specific problem of proving that 8 is a factor of ${7^{2n+1} + 1}$, showcasing the elegance and effectiveness of induction in action. Let's embark on this journey of mathematical discovery and unravel the proof step by step.

Base Case: $n = 0$

To initiate our proof by induction, the crucial first step is to establish the base case. This involves demonstrating that the statement we aim to prove holds true for the smallest value within our domain of interest. In this particular scenario, we are dealing with the set of all non-negative integers, making n = 0 the natural starting point for our base case. Our objective is to show that when n is equal to 0, the expression ${7^{2n+1} + 1}$ is indeed divisible by 8.

Substituting n = 0 into the expression, we get:

${ 7^{2(0)+1} + 1 = 7^{1} + 1 = 7 + 1 = 8 }$

The result, 8, is undeniably divisible by 8, as 8 divided by 8 yields 1 with no remainder. This straightforward calculation confirms that the statement holds true for the base case of n = 0. The significance of this step cannot be overstated, as it lays the groundwork for the rest of our inductive argument. Without a valid base case, the entire proof would crumble, as we would lack the initial domino to set off the chain reaction. Now that we have successfully established the base case, we can confidently move on to the next stage of our proof: formulating the inductive hypothesis.

The base case serves as the bedrock upon which the rest of the inductive argument is constructed. It provides the initial validation necessary to proceed with the more complex steps of the proof. By meticulously verifying the base case, we ensure that our inductive argument has a solid foundation, increasing our confidence in the overall validity of the proof. With the base case securely in place, we are well-prepared to tackle the inductive hypothesis and the inductive step, bringing us closer to our goal of proving that 8 is a factor of ${7^{2n+1} + 1}$ for all non-negative integers n. The journey of mathematical discovery is often one of careful steps, and the base case is undoubtedly a critical step in the process of inductive reasoning.

Inductive Hypothesis: Assume True for $n = k$

Having established the base case, the next pivotal step in our inductive proof is to formulate the inductive hypothesis. This is where we make a crucial assumption that will serve as the cornerstone for the remainder of our argument. The inductive hypothesis involves assuming that the statement we are trying to prove holds true for some arbitrary integer k. In our specific case, this translates to assuming that 8 is a factor of ${7^{2k+1} + 1}$ for some integer k.

Mathematically, we can express this assumption as follows:

${ 7^{2k+1} + 1 = 8m }$

where m is an integer. This equation succinctly captures the essence of our inductive hypothesis: we are assuming that the expression ${7^{2k+1} + 1}$ is a multiple of 8. The integer m simply represents the quotient when ${7^{2k+1} + 1}$ is divided by 8. The power of the inductive hypothesis lies in its ability to act as a bridge, connecting the case for n = k to the case for n = k + 1. By assuming the truth of the statement for n = k, we create a foundation upon which we can build our argument for the next value in the sequence.

The inductive hypothesis is not merely a blind assumption; it is a strategic move that allows us to leverage the power of inductive reasoning. It is akin to assuming that a particular domino in a chain will fall, setting the stage for us to demonstrate that it will indeed knock over the next domino. Without this assumption, we would be unable to link the case for n = k to the case for n = k + 1, rendering our inductive argument incomplete. The inductive hypothesis provides us with the necessary leverage to progress towards our ultimate goal of proving the statement for all non-negative integers n. With the inductive hypothesis firmly in place, we are now poised to tackle the final, and perhaps most challenging, step in our proof: the inductive step.

The inductive hypothesis is the heart of the inductive argument, providing the crucial link between successive cases. It is the assumption that allows us to build upon the base case and extend the truth of the statement to all natural numbers. By carefully formulating and applying the inductive hypothesis, we unlock the full potential of mathematical induction, enabling us to prove statements that would otherwise be intractable. As we move on to the inductive step, we will see how the inductive hypothesis is put into action, leading us to the triumphant conclusion of our proof.

Inductive Step: Prove True for $n = k + 1$

With the base case established and the inductive hypothesis assumed, the final, and often most intricate, step in our inductive proof is the inductive step. This is where we demonstrate that if the statement holds true for n = k, as we assumed in the inductive hypothesis, then it must also hold true for n = k + 1. In our specific context, this means we need to show that if 8 is a factor of ${7^{2k+1} + 1}$, then 8 must also be a factor of ${7^{2(k+1)+1} + 1}$.

Let's begin by considering the expression for n = k + 1:

${ 7^{2(k+1)+1} + 1 }$

Our goal is to manipulate this expression in such a way that we can utilize our inductive hypothesis, which states that ${7^{2k+1} + 1 = 8m}$ for some integer m. To achieve this, we can expand the exponent and rewrite the expression as follows:

${ 7^{2k+3} + 1 = 7^{2k+1+2} + 1 = 7^{2k+1} imes 7^2 + 1 = 49 imes 7^{2k+1} + 1 }$

Now, we want to introduce the term ${7^{2k+1} + 1}$, which appears in our inductive hypothesis. To do this, we can rewrite the expression by adding and subtracting 49:

${ 49 imes 7^{2k+1} + 1 = 49 imes 7^{2k+1} + 49 - 49 + 1 = 49(7^{2k+1} + 1) - 48 }$

Here, we have successfully isolated the term ${7^{2k+1} + 1}$, which we know from our inductive hypothesis is equal to 8m. Substituting this into the expression, we get:

${ 49(7^{2k+1} + 1) - 48 = 49(8m) - 48 }$

Now, we can factor out 8 from the expression:

${ 49(8m) - 48 = 8(49m - 6) }$

This final expression, 8(49m - 6), clearly demonstrates that ${7^{2(k+1)+1} + 1}$ is a multiple of 8, since it is 8 times the integer (49m - 6). This completes the inductive step. We have shown that if 8 is a factor of ${7^{2k+1} + 1}$, then 8 is also a factor of ${7^{2(k+1)+1} + 1}$.

The inductive step is the linchpin of the entire inductive argument. It establishes the crucial link between successive cases, demonstrating that the truth of the statement propagates from one value to the next. By successfully navigating the inductive step, we bridge the gap between the inductive hypothesis and the conclusion, solidifying the validity of our proof. With the base case, inductive hypothesis, and inductive step all firmly in place, we can confidently assert that our statement holds true for all non-negative integers n. The journey of mathematical induction culminates in this triumphant moment, where we demonstrate the power and elegance of this fundamental proof technique.

Conclusion: $8$ is a factor of $7^{2n+1} + 1$ for all $n$

Having meticulously navigated through the base case, inductive hypothesis, and inductive step, we now arrive at the conclusion of our proof by induction. We have successfully demonstrated that for all non-negative integers n, 8 is indeed a factor of the expression ${7^{2n+1} + 1}$. This elegant result showcases the power and effectiveness of mathematical induction as a tool for proving statements that hold across an infinite range of cases.

In summary, our proof unfolded as follows:

1. Base Case: We showed that the statement holds true for the base case of n = 0. By substituting n = 0 into the expression ${7^{2n+1} + 1}$, we obtained 8, which is clearly divisible by 8.
2. Inductive Hypothesis: We assumed that the statement is true for some arbitrary integer k. This assumption, expressed as ${7^{2k+1} + 1 = 8m}$ for some integer m, formed the cornerstone of our inductive argument.
3. Inductive Step: We demonstrated that if the statement holds true for n = k, then it must also hold true for n = k + 1. Through a series of algebraic manipulations, we showed that ${7^{2(k+1)+1} + 1}$ can be expressed as 8(49m - 6), which is a multiple of 8.

By successfully completing these three steps, we have rigorously proven that 8 is a factor of ${7^{2n+1} + 1}$ for all non-negative integers n. This result not only provides us with a specific mathematical truth but also illustrates the broader applicability of mathematical induction. This technique can be used to prove a wide array of statements across various mathematical domains, making it an indispensable tool in the mathematician's arsenal. The beauty of induction lies in its ability to transform an infinite problem into a finite one, allowing us to conquer seemingly insurmountable challenges with a systematic and logical approach.

The principle of mathematical induction is a testament to the elegance and power of mathematical reasoning. It allows us to establish truths that extend far beyond our immediate observations, providing a framework for understanding and exploring the infinite. As we conclude this proof, we celebrate not only the specific result we have obtained but also the broader power of mathematical induction to illuminate the hidden structures and patterns that underlie the mathematical universe. The journey of mathematical discovery is a continuous one, and induction serves as a reliable compass, guiding us through the vast and intricate landscape of mathematical truth.