This problem recently came up in an FP3 (old P6) textbook, and none of my 7-strong further maths class have been able to solve it
Prove by induction that:
Thanks for any help you can give; this problem has really started to irritate us!
This problem recently came up in an FP3 (old P6) textbook, and none of my 7-strong further maths class have been able to solve it
Prove by induction that:
Thanks for any help you can give; this problem has really started to irritate us!
This will only be a sketch of the proof, I will leave the demonstration of theOriginally Posted by rabywebb
truth of the base case to the reader, as well as the weasle words needed
to make this a proof by induction.
We are asked to prove:
Let (for convienience only):
and
So we now need to prove that:
Suppose this is true for then we need to prove that it is true
for .
Now:
and:
Thus to prove what is to be proven for is equivalent to proving that:
,
or:
.
Which may be simplified to:
.
To prove this (there must be many other ways) use the trig-identity for
the of a sum to obtain:
.
Now substitute the RHS of for on both the RHS and LHS of ,
on simplification these are both:
.
Which proves:
from the assumption:
.
RonL