calculus- find an explicit formula for a(n)

Question

darkigloo · Answer

$$a _{1}=\frac{ 1 }{ 2-\frac{ 1 }{ 2 }} , a _{2}=\frac{ 2 }{ 3-\frac{ 1 }{ 3}} , a _{3}=\frac{ 3 }{ 4-\frac{ 1 }{ 4}}....a _{5}$$

darkigloo · Answer

find lim as n approaches infinity of a(n)

satellite73 · Answer

is it not clear ?

satellite73 · Answer

what would $a_4$ be?  if you know that, then you know them all

darkigloo · Answer

$$a _{4}= \frac{ 4 }{ 5-\frac{ 1 }{ 5 } }$$

satellite73 · Answer

yeah ok so you got it

darkigloo · Answer

but how do i write a formula?

satellite73 · Answer

$$a_{50}=?$$

darkigloo · Answer

$$a _{50}=\frac{ 50 }{ 51- \frac{ 1 }{51 }}$$

satellite73 · Answer

sure so if you can do it with $50$ you can certainly do it with $n$ right?

satellite73 · Answer

oh btw i think you made a mistake on that one
i think what you wrote was $$a_{51}$$

satellite73 · Answer

oh no, i made a mistake, not you

darkigloo · Answer

oh i got it!

satellite73 · Answer

i mean i can tell you the answer, but visualize $n$ instead of $50$

satellite73 · Answer

what goes up, what goes down?

darkigloo · Answer

n/((n+1)-(1/(n+1)))

satellite73 · Answer

lol yeah that one

satellite73 · Answer

$$\frac{n}{(n+1)+\frac{1}{n+1}}$$

darkigloo · Answer

how do i find lim as n approaches infinity of a(n)

satellite73 · Answer

limit should be obvious

satellite73 · Answer

but i see that it is not, so lets do some number therapy first, which for some reason most people don't like i put $n=100$ which is not that big, check this http://www.wolframalpha.com/input/?i=%28100%29%2F%28101%2B1%2F101%29

satellite73 · Answer

as $n\to\infty $ $\frac{1}{n+1}\to 0$ right ?

satellite73 · Answer

so you are left with $$\frac{n}{n+1}$$ and that limit is clear 
if you want to spend a bunch of time, you could do the algebra to see what you get

satellite73 · Answer

by which i mean, get rid of the compound fraction by multiplying top and bottom by $n+1$ to get a rational expression

darkigloo · Answer

thank you

satellite73 · Answer

yw