Like what you saw?
Create FREE Account and:
 Watch all FREE content in 21 subjects(388 videos for 23 hours)
 FREE advice on how to get better grades at school from an expert
 Attend and watch FREE live webinar on useful topics
Mathematical Induction  Problem 1
Carl Horowitz
Carl Horowitz
University of Michigan
Runs his own tutoring company
Carl taught upperlevel math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
We’re now going to use mathematical induction to prove the sum of n introduced is equal to n times n plus 1 over 2.
Our first step is to show it works for n equals 1. Basically we’re assuming the first integer, obviously the sum of the first integer is just going to be 1. But we want to make sure that our formula holds. So when we plug in one here we get 1 times 1 plus 1, 1 times 2 is 2, over 2 which is equal to 1, so we have 1 is equal to 1, that works.
We now assume that it works for some arbitrary value k. What that tells us is we end up with 1 plus 2 plus 3 plus so on and so forth, all the way up to k, is going to be equal to k times it’s going to be k plus 1 over 2. So then sing this fact we want to show that it works for k plus 1. So what we can do is show it works for k plus 1.
What we’re looking for is the sum of 1 plus 2 plus 3 plus k plus k plus 1. What we have is this piece we already know to be k times k plus 1 over 2, because all we did is we added k plus to this side we therefore we add k plus 1 to that side as well. All we have to do in this case is to get a common denominator. In this case this is going to be 2. Multiply this by 2 over 2 and we’re left with, let’s get a different color so we can distinguish it a little bit more. Foiling this out k² plus k over 2 plus 2k plus 2 over 2. Combining like terms we end up with k² plus 3k plus over 2, which hopefully you can see factors down into k plus 1 times k plus 2 over 2.
So that is on one side of our equation. If we were to just add k plus 1 which is the number we added to this side, we would end up with k plus 1 times k plus 2 over 2. We want to prove that this statement is the same as this statement over here. For this statement we can just see our formula.
The sum of the first n terms is simply n times n plus over 2. So the sum of the first k plus 1 terms is simply our n 1 to n so simply k plus 1. We added one to that over here so that turns into k plus 2 over 2.
So basically by assuming, by proving it for one, assuming it works for k, we were then able to manipulate this equation to get the equation to then work for k plus 1, thus proving that this statement is true by mathematical induction.
Please enter your name.
Are you sure you want to delete this comment?
Carl Horowitz
B.S. in Mathematics University of Michigan
He knows how to make difficult math concepts easy for everyone to understand. He speaks at a steady pace and his stepbystep explanations are easy to follow.
i love you you are the best, ive spent 3 hours trying to understand probability and this is making sense now finally”
BRIGHTSTORM IS A REVOLUTION !!!”
because of you i ve got a 100/100 in my test thanks”
Get Peer Support on User Forum
Peer helping is a great way to learn. Join your peers to ask & answer questions and share ideas.
Concept (1)
Sample Problems (6)
Need help with a problem?
Watch expert teachers solve similar problems.

Mathematical Induction
Problem 1 6,231 viewsProve:
1 + 2 + 3 +....... n = n(n + 1) 2 
Mathematical Induction
Problem 2 5,118 viewsProve:
1 + 3 + 5 +.....(2n  1) = n² 
Mathematical Induction
Problem 3 792 views 
Mathematical Induction
Problem 4 684 views 
Mathematical Induction
Problem 5 703 views 
Mathematical Induction
Problem 6 742 views
Comments (1)
Please Sign in or Sign up to add your comment.
·
Delete
Liên · 10 months ago
I'm confused to where you got "k+k+1" at 1:34.