a1=1
a_n+1=a_n*1/2
how to prove that
a_n+1 -a_n<0
?
i tried by induction
a1>0
suppose a_n>0
a_n+1 -a_n>0
-a_n*1/2<0
so its monotonicly desendant correct?
It is a little 'surpising' the fact that You have a total of 973 posts and none of them is written in Latex!... the question is not minor because what You have written can cause misundestanding!...
If the 'recursive relation' is...
$\displaystyle \displaystyle a_{n+1}= \sqrt{a_{n}}$ , $\displaystyle a_{1}=1$ (1)
... then is $\displaystyle \forall n\ \ a_{n}=1$ and the sequence is neither increasing nor decreasing...
If the 'recursive relation' is...
$\displaystyle \displaystyle a_{n}+1= \sqrt{a_{n}}$ , $\displaystyle a_{1}=1$ (2)
... then is...
$\displaystyle \displaystyle \Delta_{n} = a_{n-1}-a_{n} = \sqrt{a_{n}} - a_{n}- 1$ (3)
... and $\displaystyle \forall a_{n}>0$ the (3) is negative so that the sequence is decreasing and it finishes at the n for which is $\displaystyle a_{n}<0$...
There are some other 'possible interpretations' but in any case the 'advatages' of the use of Latex are fully evident!...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
I would have interpreted that as $\displaystyle a_{n+1}= \frac{a_n}{2}$!
To prove, by induction that this is a strictly decreasing sequence, that is, that $\displaystyle a_{n+1}< a_n$ for all n, note that $\displaystyle a_1= 1$ and that $\displaystyle a_2= \frac{1}{2}$ so the statement is true for n=1.
Assume that $\displaystyle a_{k+1}< a_k$ for some k. Then $\displaystyle a_{k+2}= \frac{a_{k+1}}{2}$ but since $\displaystyle a_{k+1}< a_k$, $\displaystyle \frac{a_{k+1}}{2}< \frac{a_k}{2}= a_{k+1}$. That is, $\displaystyle a_{k+2}< a_{k+1}$ and we are done.
If it is, in fact, $\displaystyle a_{n+1}= \sqrt{a_n}$, then the sequence is neither increasing nor decreasing- $\displaystyle a_n= 1$ for all n!