Proof Prove each formula by mathematical induction. (You may need to review the method of proof

Proof Prove each formula by mathematical induction. (You may...

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the validity of a statement for all natural numbers. The method involves proving a base case for the smallest value (usually 1) and then showing that if the statement holds for an arbitrary case n, it must also hold for the case n+1. This step-by-step approach ensures that the statement is true for every natural number, by creating a domino effect where each case leads to the next.

Base Case

The base case is the initial step in an induction proof where the statement is verified for the smallest or initial value of the variable, typically n = 1. Establishing the base case is crucial because it starts the induction process. If the base case is not true, the entire inductive argument fails, as there is no foundation from which the induction can proceed.

Inductive Hypothesis

The inductive hypothesis is the assumption step in an induction proof where it is assumed that the statement holds for some arbitrary natural number n. This hypothesis is not assumed to be a proven fact beyond n—it is temporarily held true to facilitate proving the next step. The hypothesis is the linchpin that connects the base case to the inductive step, enabling one to extend the validity from one case to the next.

Inductive Step

The inductive step is the component of an induction proof in which one proves that the validity of the statement for n (assumed through the inductive hypothesis) implies its validity for n+1. This often involves algebraic manipulation and logical deduction to show that if the statement holds for a general case, the subsequent case cannot fail. Successfully proving the inductive step completes the proof mechanism.

Summation of Sequences

Summation of sequences involves adding a series of numbers defined by a specific rule or pattern. In many mathematical contexts, particularly in induction proofs, formulas are derived or verified to represent the sum of the first n terms of a sequence. Understanding summation formulas is essential for working with series and is a common topic in algebra and precalculus, often requiring manipulation of expressions to confirm the correctness of a proposed formula.

Transcript

00:01 Alright, so we want in half the formula for the sum of i equals 1 up to n, the sum from i equals 1 up to n of i.

00:21 This formula is equal to n times n plus 1 divided by 2.

00:31 So you want to do, i want to prove that that is true.

00:40 So the proof you can do a proof by induction.

00:44 So by induction, basically need to do two things.

00:55 So first we prove on base case, which generally speaking is like, prove that the formula holds for n equals 1 or 0 or 2, some end is more.

01:16 So this formula, let's call that formula star, so prove that star holds, let's start that n equals, one for n equals one so for n equals one on this side well it's an equality so you have to prove that the left hand side agrees with the right hand side so for n equals 1 you have all the sum from i equals 1 up to 1 of 1 is just of i is just 1 so that is the left hand side left hand side and if you plug in and equal one for the right hand side is that equal right and side is that equal to one plus one times one plus one or two so yeah that is two or two so it is equal to one times one it is true so you have the base case now we need to prove a second thing for the profile induction you need to prove that the formula if the formula starts holds for p, this fact implies that the whole formula star starr holds for for, well the formula holds for not p for n holds that n, this fact implies that the formula holds for n plus 1.

03:14 So if you have this implication together with the base case, this this this proofs by induction that is the principle of induction that the formula holds for any and for any so all the base case is done so let's do the second part prove that if it holds for n it holds for n plus 1 so as a base case so our induction hypothesis is that if you but the formula holds for n so that means if you add i from i equals 1 up to n you're going to get n times n plus 1 halves what is going to be the sum up to n plus 1 from i equals 1 so the sum can be written as well the sum up to n of i plus the n plus the n plus the n plus one time so it's going to be this plus n plus 1 but using the induction hypothesis this here all this is that so we can just replace what is that we would have that that is equal to n times n plus 1 divided by 2 and that is we're adding n plus 1 to that plus n plus 1 so to two other decent in terms of all that we need to put a common denominator of 2 so let's multiply by 2 on top and on bottom so we have that or we're going to have 2m plus 1 plus n times m plus 1 which is this term over 2 over 2 and so and so if we go, let's try and see what can we factor.

06:09 So here we have an n squared plus n...

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