Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 25

Capitolo 9, Problema 25

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n² - n.

Guida verificata passo dopo passo

First, identify the statement to prove using mathematical induction: For every positive integer $n$, $2$ is a factor of $n^{2} - n$. This means we want to show that $n^{2} - n$ is divisible by $2$ for all $n \geq 1$.

Start with the base case: check if the statement holds for $n = 1$. Substitute $n = 1$ into the expression $n^{2} - n$ and verify if the result is divisible by $2$.

Assume the statement is true for some positive integer $k$, i.e., assume $2$ divides $k^{2} - k$. This is called the induction hypothesis.

Using the induction hypothesis, prove the statement for $n = k + 1$. Substitute $k + 1$ into the expression to get $(k + 1)^{2} - (k + 1)$ and simplify it.

Show that the expression for $n = k + 1$ can be written as a sum of terms, one of which is divisible by $2$ by the induction hypothesis, and the other terms are also divisible by $2$. This will complete the induction step and prove the statement for all positive integers $n$.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Mathematical Induction

Mathematical induction is a proof technique used to show that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1.

Video consigliato:

05:17

Types of Slope

Divisibility and Factors

Understanding divisibility means recognizing when one integer is a factor of another. In this problem, showing that 2 is a factor of n² - n means proving that n² - n is always even, i.e., divisible by 2, for every positive integer n.

Video consigliato:

07:30

Introduction to Factoring Polynomials

Algebraic Manipulation of Expressions

Algebraic manipulation involves rewriting expressions to reveal properties like factors or divisibility. For n² - n, factoring it as n(n - 1) helps identify that the product of two consecutive integers is always even, which is key to the proof.

Video consigliato:

05:09

Introduction to Algebraic Expressions

Pratica correlata

Domanda del libro di testo

Evaluate each expression.

$\frac{_{7} C_{3}}{_{5} C_{4}} - \frac{98!}{96!}$

590

views

Domanda del libro di testo

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)⁵

667

views

Domanda del libro di testo

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 12 terms of the geometric sequence: 2, 6, 18, 54, ...

1112

views

Domanda del libro di testo

In Exercises 23–28, evaluate each factorial expression. 20!/2!18!

637

views

Domanda del libro di testo

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)⁵

706

views

Domanda del libro di testo

Evaluate each factorial expression. 16!/2!14!