Prove or disprove (expansion sequences)?

Question

Question:
http://img401.imageshack.us/img401/7609/asdfec4.gif

on the right whats blurry is {p(n)}

and then for all integers n greater than or equal to 0

I have no idea what it means please help!

Answer 1

By induction.

n=0: p0/q0=a0 -- true
n=1: p1/q1=a0+1/a1 -- true

Let the formula be true for n
pk/qk
= (akp[k-1] + p[k-2]) / (akq[k-1] + q[k-2])
= (aka[k-1]p[k-2] + akp[k-3] + p[k-2]) / (aka[k-1]q[k-2] + akq[k-3] + q[k-2])
= ((a[k-1] + 1/ak)p[k-2] + p[k-3]) / ((a[k-1] + 1/ak)q[k-2] + q[k-3])
= p[k-1]'/q[k-1]'
where ps' and qs' are calculated for the following sequence
a0'=a0, a1'=a1, ..., a[k-2]'=a[k-2], a[k-1]'=a[k-1]+1/ak
Then,
p0'=p0, ..., p[k-2]'=p[k-2], p[k-1]'=(a[k-1]+1/ak)p[k-2] + p[k-3]
q0'=q0, ..., q[k-2]'=q[k-2], q[k-1]'=(a[k-1]+1/ak)q[k-2] + q[k-3]

For this sequence, by induction assumption for n=k-1, we have
p'[k-1]/q'[k-1] = a0 + 1/(a1 + 1/(a2 + ... + 1/(a[k-1] + 1/ak)...))

Answer 2

So you have a set of numbers a1, a2, a3, a4..., and they're used to define two sequences. They want you to prove or disprove that the expression they give for pn / qn is correct for any n. You could try proving this by induction.

Answer 3

Einstein's Relative theory is much shorter. Ok...let me put it this way: We know the Big Bang thing happened....my bad...is CONTINUING to happen and the Universe is EXPANDING. Question is: where?

Answer that and you snag a Nobel.