Axiom of Induction Uh Oh
Let's look at this sequence...
1/[1*2] + 1/[2*3] + 1/[3*4] + ....
Get the pattern?
1/[n(n+1)] = n/[n+1]
So we have...
1/[1*2] + 1/[2*3] + 1/[3*4] + .... + 1/[n(n+1)] = n/[n+1]
I'm supposed to show that the statement holds for all positive integers n.
This will require the use of mathematical induction.
I haven't really learned this yet so be patient. I know I have to show that when n=1 the statement is true.
So we have...
n=1 means that we just do the first term (left side), which is 1/(1*2) = 1/2 or 0.5. The right hand side is 1/(1+1) = 1/2 = 0.5 SAME YAY so far so good!
But then it wants me to show that they n=k and then it wants me to show that n=k+1.
Can someone walk me through these steps?
Yes my algebra is beyond bad, but could you explain what you did here ?:
This one is fine
How did you get this?
Originally Posted by Jones
First, we must assume that the statement is correct for some positive integer n. That is, is assumed to be correct.
According to the principle of induction, it will be correct for all positive integers if we prove that, given the assumption that it is correct for some positive integer n, it is also correct for n+1. That is to say, we now need to prove that the following equation:
However, from the assumption we have, we know that
So we can substitute this expression from the LHS of (*), and then we get:
as the LHS.
The RHS of (*) stays . Now we need to prove both sides are equal. Let's play with the left hand side:
So the LHS of (*) is equal to . But so is the RHS, so we conclude that (*) holds and therefore the equation is correct.