Hey guys,
ive got a question to solve:
It goes like this:
Using mathematical Induction, prove that
SUM(r=1->n) of (r+1)2^r-1 = n(2^n)
Now the basic step is easy so im not gonna bother typing it here now but later on:
I get that:
(Let P(k) be true so
Sum(r=1->k) of (r+1)2^r-1 = k(2^k)
Now prove that P(k+1) is true:
Sum(r=1->k+1) of (r+1)2^r-1 = (k+1)(2^(k+1))
The Lefthandside of the equation above says:
Sum(r=1->k+1) of (r+1)2^r-1
We can expand this to be
Sum(r=1->k) of (r+1)2^r-1 + ((k+1)+1)2^(k+1)-1
Using our assumption of the beginnig we can say now that:
k(2^k) + ((k+1)+1)2^(k+1)-1 must equal this if i havenīt done any mistakes: (k+1)(2^(k+1)), doesnīt it.
But somehow i canīt get that,
if some could just show me the last step then iīd be fine with that.
Hope i`ve expressed my self correctly in that question. (Couldnīt really get used to the commands quick menu; sorry)
Cheers guys
This is how far I just got a minute ago as well;
but I am stuck there and canīt get it to be (k+1)(2^(k+1)).
Would it be fine ich I just state an example, where I replace the k with a fixed value!?
(2k+2)2^k has to be fromed to look like that: (k+1)(2^(k+1))
Thatīs what I thought as well, it is just not the point of the induction.
I donīt get that step you have done
(2k+2)2^k =2.2^k(k+1)=(k+1)2^(k+1)
I know i could just write that down, but i want to understand it as well.
Can you explain it in more detail please?
Would be really nice, thank you
EDIT:
No i got it now.
The last step was fairly easy but i just didnīt really know what that dot between the two twos was meant to be.
Anyway thank you!!!