A solution without using induction!
Let
Multiplying by :
Now, using the identity :
Splitting the R.H.S into two separate summations, we will get:
Or:
The above is telescopic series:
Hence:
first up, sorry if this is the wrong forum to post but i was unsure where to post it under.
i am trying to prove that:
(ment to read sum( cos(ku), k=1 to n))
my induction assumption is that:
and so i am trying to reduce:
to become
this is where i am stuck. any help would be appreciated greatly
A solution without using induction!
Let
Multiplying by :
Now, using the identity :
Splitting the R.H.S into two separate summations, we will get:
Or:
The above is telescopic series:
Hence:
First of all, let's write this in a form that is readable...
You're asked to prove that
.
Base step :
.
.
Inductive step - Assume the statement is true for .
So we assume .
Now we need to show that the statement is true for , i.e. that
.
.
Now I'm a little stuck too, will keep thinking about it...
P(n)
P(n+1)
Proof
If P(n) is valid, this will equal
Hence, the question is
Expanding the rightmost term
The inter-term relationship is proven,
hence the trail of gunpowder has been placed all the way to that bank.
"What bank?"
That bank at infinity!
All that's required is to prove the equality is true for n=1..
and since
and using
the result follows