Help with proof by induction


by knowledgeSeeker
Tags: induction, proof
knowledgeSeeker
knowledgeSeeker is offline
#1
Mar2-05, 11:22 PM
P: 2
Need help with proof by mathematical induction that (1/(1*2)) + (1/(2*3)) + ... + (1/(n(n+1)) = (n/(n+ 1)) for all integers n >= 1.

Basis step: for n = 1: (1/(1*2)) = 1/2 and (1/(1+1) = 1/2, hence property is true for n = 1.

Inductive step: want to show that for alll integers k >= 1, if n = k is true then n = k + 1 is true. How do I prove? Believe I want to show (1/(1*2)) + (1/(2*3)) + [1/((k+1)((k+1)+1)] = [(k + 1)/((k+1) + 1)], but how??

Thank you for any suggestions.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
gnome
gnome is offline
#2
Mar3-05, 12:16 AM
P: 1,047
So you have shown that P(1) is true. Now you want to show that if you assume that P(k) is true, it follows that P(k+1) is true. So first write the expression for P(k), which you assume to be true. Then add the next number in the series (to both sides), and see if you can rearrange the expression on the right side into the form that you are trying to prove.
HallsofIvy
HallsofIvy is offline
#3
Mar3-05, 07:26 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,898
Let Sk= 1/(1*2)+ 1/(2*3)+ ...+ 1/(k(k+1)), the sum for n= k
Then S(k+1)= 1/(1*2)+ ...+ 1/(k)(k+1)+ 1/((k+1)((k+1)+1)= Sk+ 1/((k+1)(k+2))

By your "induction hypothesis", Sk= k/(k+1).

What is k/(k+1)+ 1/((k+1)(k+2)) ?

knowledgeSeeker
knowledgeSeeker is offline
#4
Mar3-05, 09:28 AM
P: 2

Help with proof by induction


Thank you. Proved both sides = (k+1)/(k+2). Hence, true for n = k +1 and since both basis and inductive steps true, true for all n >= 1.


Register to reply

Related Discussions
proof by induction: help General Math 4
Induction Proof Set Theory, Logic, Probability, Statistics 3
Proof by induction Calculus & Beyond Homework 4
another induction proof Precalculus Mathematics Homework 5
Proof by Induction Calculus & Beyond Homework 5