induction q (1 Viewer)

[polythenepam]

just a small part of an induction question.. (part b only)
u[1]=1 and u[n]=(2u[n-1])^1/2 (..where the [n] and [n+1] etc denotes the term)
a) show u[n]<2 for n>or equal to 1
b) deduce that u[n+1]>u[n] for n>or equal to 1
ive done a) i just dont know how to do b).. some help please?

 

[011]

just a small part of an induction question.. (part b only)
u[1]=1 and u[n]=(2u[n-1])^1/2 (..where the [n] and [n+1] etc denotes the term)
a) show u[n]<2 for n>or equal to 1
b) deduce that u[n+1]>u[n] for n>or equal to 1
ive done a) i just dont know how to do b).. some help please?

There might be some theoretical quirks not ironed out, which I suppose could be alleviated if you initially state u[n] > 0 for n>=1, but this is what I came up with:

We know u[n] < 2, n >=1
u[n]-1 < 1
(u[n]-1)^2 < 1 (square)
(u[n])^2 - 2u[n] < 0
So we have 2u[n] > (u[n])^2
(2u[n])^1/2 > u[n] (square root)
Substituting into the formula, LHS = u[n+1]
Thus we have the condition u[n+1] > u[n]

 

[polythenepam]

ohhhkay i see now.. thanx heaps..!