   Chapter 2.2, Problem 50E

Chapter
Section
Textbook Problem

Show that if the statement 1 + 2 + 3 +   ... + n = n ( n + 1 ) 2 + 2 is assumed to be true for n = k , the same equation can be proved to be true for n = k + 1 . Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?

To determine

Whether the statement “ 1+2+3++n=n(n+1)2+2 is assumed to be true for n=k,” then it is true for n=k+1 and explain why the statement is not true for all positive integers n.

Explanation

Given information:

The given statement is, “ 1+2+3++n=n(n+1)2+2 ”is true for n=k.

Formula used:

For all positive integers n, the statement Pn is true if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

Proof:

Let Pn be the statement, “ 1+2+3++n=n(n+1)2+2

Assume that Pk is true.

1+2+3++k=k(k+1)2+2

For n=k+1

By induction hypothesis, 1+2+3++k=k(k+1)2+2

Adding k+1 on both sides,

1+2+3++k+(k+1)=k(k+1)2+2+(k+1)

Simplify R

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Place the following set of n = 20 scores in a frequency distribution table.

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Find the general indefinite integral. t(t2+3t+2)dt

Single Variable Calculus: Early Transcendentals, Volume I

True or False: n=1n+n3n2/3+n3/2+1 is a convergent series.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

True or False: converges.

Study Guide for Stewart's Multivariable Calculus, 8th 