Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 31

Chapter 9, Problem 31

Use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Verified step by step guidance

Identify the statement to prove using mathematical induction: For every positive integer $n$, the inequality $n + 2 > n$ holds.

Base Case: Verify the statement for $n = 1$. Substitute $n = 1$ into the inequality to check if $1 + 2 > 1$ is true.

Inductive Hypothesis: Assume the statement is true for some positive integer $k$, that is, assume $k + 2 > k$ holds.

Inductive Step: Using the inductive hypothesis, prove the statement for $k + 1$. Show that $(k + 1) + 2 > k + 1$ is true.

Conclude that since the base case is true and the inductive step holds, by mathematical induction, the inequality $n + 2 > n$ is true for every positive integer $n$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually for n=1) and then proving the inductive step, where assuming the statement is true for n=k leads to it being true for n=k+1.

Base Case Verification

The base case is the initial step in induction where the statement is verified for the smallest positive integer, often n=1. This step confirms the statement holds at the starting point, providing a foundation for the inductive step.

Inductive Step

The inductive step requires assuming the statement is true for an arbitrary positive integer n=k (inductive hypothesis) and then proving it is true for n=k+1. This step shows the property holds for the next integer, completing the induction process.

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Use the Fundamental Counting Principle to solve Exercises 29–40. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

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Textbook Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for a_n to find a₂₀, the 20th term of the sequence. a_n = a_n-1 +3, a₁ = 4

Find each indicated sum. $\sum_{i = 1}^{4} 2 i^{2}$

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Textbook Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)⁸

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