PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 9, Problem 72RE
Solution

To prove: The given statement is true or not for any positive integer using mathematical induction.

Given information:

The given statement is 12+23+34++nn+1=nn+1n+23 .

Formula used :

Follow the steps to prove that the statement is true for any positive integer.

Anchor step: Prove that any statement Pn is true for n=1 .

Inductive hypothesis: Assume Pn is true for n=k .

Inductive Step: Prove that Pn is true for n=k+1 .

Explanation:

Conjecture:

Take Pn to be statement 12+23+34++nn+1=nn+1n+23 .

Anchor step:

Prove that Pn is true for n=1 .

For n=1 ,

  12=11+11+2312=12332=2

Inductive hypothesis:

Assume Pn is true for n=k . Replace n with k in Pn .

  12+23+34++kk+1=kk+1k+23

Inductive Step:

Prove that Pn is true for n=k+1 .

This means it is needed to be proved that Pk+1 must be true.

Proceed with the inductive hypothesis.

  12+23+34++kk+1=kk+1k+23

Add k+1th term on both sides and simplify the right-hand side.

  1+3+6++kk+12+k+1k+2=kk+1k+23+k+1k+2=k+1k+2k3+1=k+1k+2k+33

It can be written as:

  k+1k+2k+33=k+1k+1+1k+1+23

The statement is same as the statement Pk+1 , so Pk+1 is true.

So, if Pn is true for n=k , then it is also true for n=k+1 .

Conclusion:

As Pn is true for n=1 and Pk further implies Pk+1 , this means Pn is true for n=2 , n=3 and so on.

By mathematical induction, 12+23+34++nn+1=nn+1n+23 is true for all positive integers n .

Chapter 9 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 9.1 - Prob. 1ECh. 9.1 - Prob. 2ECh. 9.1 - Prob. 3ECh. 9.1 - Prob. 4ECh. 9.1 - Prob. 5ECh. 9.1 - Prob. 6ECh. 9.1 - Prob. 7ECh. 9.1 - Prob. 8ECh. 9.1 - Prob. 9ECh. 9.1 - Prob. 10ECh. 9.1 - Prob. 11ECh. 9.1 - Prob. 12ECh. 9.1 - Prob. 13ECh. 9.1 - Prob. 14ECh. 9.1 - Prob. 15ECh. 9.1 - Prob. 16ECh. 9.1 - Prob. 17ECh. 9.1 - Prob. 18ECh. 9.1 - Prob. 19ECh. 9.1 - Prob. 20ECh. 9.1 - Prob. 21ECh. 9.1 - Prob. 22ECh. 9.1 - Prob. 23ECh. 9.1 - Prob. 24ECh. 9.1 - Prob. 25ECh. 9.1 - Prob. 26ECh. 9.1 - Prob. 27ECh. 9.1 - Prob. 28ECh. 9.1 - Prob. 29ECh. 9.1 - Prob. 30ECh. 9.1 - Prob. 31ECh. 9.1 - Prob. 32ECh. 9.1 - Prob. 33ECh. 9.1 - Prob. 34ECh. 9.1 - Prob. 35ECh. 9.1 - Prob. 36ECh. 9.1 - Prob. 37ECh. 9.1 - Prob. 38ECh. 9.1 - Prob. 39ECh. 9.1 - Prob. 40ECh. 9.1 - Prob. 41ECh. 9.1 - Prob. 42ECh. 9.1 - Prob. 43ECh. 9.1 - Prob. 44ECh. 9.1 - Prob. 45ECh. 9.1 - Prob. 46ECh. 9.1 - Prob. 47ECh. 9.1 - Prob. 48ECh. 9.1 - Prob. 49ECh. 9.1 - Prob. 50ECh. 9.1 - Prob. 51ECh. 9.1 - Prob. 52ECh. 9.1 - Prob. 53ECh. 9.1 - Prob. 54ECh. 9.1 - Prob. 55ECh. 9.1 - Prob. 56ECh. 9.1 - Prob. 57ECh. 9.1 - Prob. 58ECh. 9.2 - Prob. 1QRCh. 9.2 - Prob. 2QRCh. 9.2 - Prob. 3QRCh. 9.2 - Prob. 4QRCh. 9.2 - Prob. 5QRCh. 9.2 - Prob. 6QRCh. 9.2 - Prob. 7QRCh. 9.2 - Prob. 8QRCh. 9.2 - Prob. 9QRCh. 9.2 - Prob. 10QRCh. 9.2 - Prob. 1ECh. 9.2 - Prob. 2ECh. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - Prob. 5ECh. 9.2 - Prob. 6ECh. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - Prob. 10ECh. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Prob. 15ECh. 9.2 - Prob. 16ECh. 9.2 - Prob. 17ECh. 9.2 - Prob. 18ECh. 9.2 - Prob. 19ECh. 9.2 - Prob. 20ECh. 9.2 - Prob. 21ECh. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - Prob. 24ECh. 9.2 - Prob. 25ECh. 9.2 - Prob. 26ECh. 9.2 - Prob. 27ECh. 9.2 - Prob. 28ECh. 9.2 - Prob. 29ECh. 9.2 - Prob. 30ECh. 9.2 - Prob. 31ECh. 9.2 - Prob. 32ECh. 9.2 - Prob. 33ECh. 9.2 - Prob. 34ECh. 9.2 - Prob. 35ECh. 9.2 - Prob. 36ECh. 9.2 - Prob. 37ECh. 9.2 - Prob. 38ECh. 9.2 - Prob. 39ECh. 9.2 - Prob. 40ECh. 9.2 - Prob. 41ECh. 9.2 - Prob. 42ECh. 9.2 - Prob. 43ECh. 9.2 - Prob. 44ECh. 9.2 - Prob. 45ECh. 9.3 - Prob. 1QRCh. 9.3 - Prob. 2QRCh. 9.3 - Prob. 3QRCh. 9.3 - Prob. 4QRCh. 9.3 - Prob. 5QRCh. 9.3 - Prob. 6QRCh. 9.3 - Prob. 7QRCh. 9.3 - Prob. 8QRCh. 9.3 - Prob. 9QRCh. 9.3 - Prob. 10QRCh. 9.3 - Prob. 1ECh. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - Prob. 4ECh. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - Prob. 11ECh. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - Prob. 15ECh. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Prob. 19ECh. 9.3 - Prob. 20ECh. 9.3 - Prob. 21ECh. 9.3 - Prob. 22ECh. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - Prob. 27ECh. 9.3 - Prob. 28ECh. 9.3 - Prob. 29ECh. 9.3 - Prob. 30ECh. 9.3 - Prob. 31ECh. 9.3 - Prob. 32ECh. 9.3 - Prob. 33ECh. 9.3 - Prob. 34ECh. 9.3 - Prob. 35ECh. 9.3 - Prob. 36ECh. 9.3 - Prob. 37ECh. 9.3 - Prob. 38ECh. 9.3 - Prob. 39ECh. 9.3 - Prob. 40ECh. 9.3 - Prob. 41ECh. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.3 - Prob. 44ECh. 9.3 - Prob. 45ECh. 9.3 - Prob. 46ECh. 9.3 - Prob. 47ECh. 9.3 - Prob. 48ECh. 9.3 - Prob. 49ECh. 9.3 - Prob. 50ECh. 9.3 - Prob. 51ECh. 9.3 - Prob. 52ECh. 9.3 - Prob. 53ECh. 9.3 - Prob. 54ECh. 9.3 - Prob. 55ECh. 9.3 - Prob. 56ECh. 9.4 - Prob. 1QRCh. 9.4 - Prob. 2QRCh. 9.4 - Prob. 3QRCh. 9.4 - Prob. 4QRCh. 9.4 - Prob. 5QRCh. 9.4 - Prob. 6QRCh. 9.4 - Prob. 7QRCh. 9.4 - Prob. 8QRCh. 9.4 - Prob. 9QRCh. 9.4 - Prob. 10QRCh. 9.4 - Prob. 1ECh. 9.4 - Prob. 2ECh. 9.4 - Prob. 3ECh. 9.4 - Prob. 4ECh. 9.4 - Prob. 5ECh. 9.4 - Prob. 6ECh. 9.4 - Prob. 7ECh. 9.4 - Prob. 8ECh. 9.4 - Prob. 9ECh. 9.4 - Prob. 10ECh. 9.4 - Prob. 11ECh. 9.4 - Prob. 12ECh. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - Prob. 15ECh. 9.4 - Prob. 16ECh. 9.4 - Prob. 17ECh. 9.4 - Prob. 18ECh. 9.4 - Prob. 19ECh. 9.4 - Prob. 20ECh. 9.4 - Prob. 21ECh. 9.4 - Prob. 22ECh. 9.4 - Prob. 23ECh. 9.4 - Prob. 24ECh. 9.4 - Prob. 25ECh. 9.4 - Prob. 26ECh. 9.4 - Prob. 27ECh. 9.4 - Prob. 28ECh. 9.4 - Prob. 29ECh. 9.4 - Prob. 30ECh. 9.4 - Prob. 31ECh. 9.4 - Prob. 32ECh. 9.4 - Prob. 33ECh. 9.4 - Prob. 34ECh. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.4 - Prob. 37ECh. 9.4 - Prob. 38ECh. 9.4 - Prob. 39ECh. 9.4 - Prob. 40ECh. 9.4 - Prob. 41ECh. 9.4 - Prob. 42ECh. 9.4 - Prob. 43ECh. 9.4 - Prob. 44ECh. 9.4 - Prob. 45ECh. 9.4 - Prob. 46ECh. 9.4 - Prob. 47ECh. 9.4 - Prob. 48ECh. 9.4 - Prob. 49ECh. 9.4 - Prob. 50ECh. 9.4 - Prob. 51ECh. 9.4 - Prob. 52ECh. 9.5 - Prob. 1QRCh. 9.5 - Prob. 2QRCh. 9.5 - Prob. 3QRCh. 9.5 - Prob. 4QRCh. 9.5 - Prob. 5QRCh. 9.5 - Prob. 6QRCh. 9.5 - Prob. 7QRCh. 9.5 - Prob. 8QRCh. 9.5 - Prob. 9QRCh. 9.5 - Prob. 10QRCh. 9.5 - Prob. 1ECh. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Prob. 4ECh. 9.5 - Prob. 5ECh. 9.5 - Prob. 6ECh. 9.5 - Prob. 7ECh. 9.5 - Prob. 8ECh. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - Prob. 11ECh. 9.5 - Prob. 12ECh. 9.5 - Prob. 13ECh. 9.5 - Prob. 14ECh. 9.5 - Prob. 15ECh. 9.5 - Prob. 16ECh. 9.5 - Prob. 17ECh. 9.5 - Prob. 18ECh. 9.5 - Prob. 19ECh. 9.5 - Prob. 20ECh. 9.5 - Prob. 21ECh. 9.5 - Prob. 22ECh. 9.5 - Prob. 23ECh. 9.5 - Prob. 24ECh. 9.5 - Prob. 25ECh. 9.5 - Prob. 26ECh. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.5 - Prob. 29ECh. 9.5 - Prob. 30ECh. 9.5 - Prob. 31ECh. 9.5 - Prob. 32ECh. 9.5 - Prob. 33ECh. 9.5 - Prob. 34ECh. 9.5 - Prob. 35ECh. 9.5 - Prob. 36ECh. 9.5 - Prob. 37ECh. 9.5 - Prob. 38ECh. 9.5 - Prob. 39ECh. 9.5 - Prob. 40ECh. 9.5 - Prob. 41ECh. 9.5 - Prob. 42ECh. 9.5 - Prob. 43ECh. 9.5 - Prob. 44ECh. 9.5 - Prob. 45ECh. 9.5 - Prob. 46ECh. 9.5 - Prob. 47ECh. 9.5 - Prob. 48ECh. 9.5 - Prob. 49ECh. 9.5 - Prob. 50ECh. 9.5 - Prob. 51ECh. 9.5 - Prob. 52ECh. 9.5 - Prob. 53ECh. 9 - Prob. 1RECh. 9 - Prob. 2RECh. 9 - Prob. 3RECh. 9 - Prob. 4RECh. 9 - Prob. 5RECh. 9 - Prob. 6RECh. 9 - Prob. 7RECh. 9 - Prob. 8RECh. 9 - Prob. 9RECh. 9 - Prob. 10RECh. 9 - Prob. 11RECh. 9 - Prob. 12RECh. 9 - Prob. 13RECh. 9 - Prob. 14RECh. 9 - Prob. 15RECh. 9 - Prob. 16RECh. 9 - Prob. 17RECh. 9 - Prob. 18RECh. 9 - Prob. 19RECh. 9 - Prob. 20RECh. 9 - Prob. 21RECh. 9 - Prob. 22RECh. 9 - Prob. 23RECh. 9 - Prob. 24RECh. 9 - Prob. 25RECh. 9 - Prob. 26RECh. 9 - Prob. 27RECh. 9 - Prob. 28RECh. 9 - Prob. 29RECh. 9 - Prob. 30RECh. 9 - Prob. 31RECh. 9 - Prob. 32RECh. 9 - Prob. 33RECh. 9 - Prob. 34RECh. 9 - Prob. 35RECh. 9 - Prob. 36RECh. 9 - Prob. 37RECh. 9 - Prob. 38RECh. 9 - Prob. 39RECh. 9 - Prob. 40RECh. 9 - Prob. 41RECh. 9 - Prob. 42RECh. 9 - Prob. 43RECh. 9 - Prob. 44RECh. 9 - Prob. 45RECh. 9 - Prob. 46RECh. 9 - Prob. 47RECh. 9 - Prob. 48RECh. 9 - Prob. 49RECh. 9 - Prob. 50RECh. 9 - Prob. 51RECh. 9 - Prob. 52RECh. 9 - Prob. 53RECh. 9 - Prob. 54RECh. 9 - Prob. 55RECh. 9 - Prob. 56RECh. 9 - Prob. 57RECh. 9 - Prob. 58RECh. 9 - Prob. 59RECh. 9 - Prob. 60RECh. 9 - Prob. 61RECh. 9 - Prob. 62RECh. 9 - Prob. 63RECh. 9 - Prob. 64RECh. 9 - Prob. 65RECh. 9 - Prob. 66RECh. 9 - Prob. 67RECh. 9 - Prob. 68RECh. 9 - Prob. 69RECh. 9 - Prob. 70RECh. 9 - Prob. 71RECh. 9 - Prob. 72RECh. 9 - Prob. 73RECh. 9 - Prob. 74RECh. 9 - Prob. 75RECh. 9 - Prob. 76RE
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