Question 872658
{{{8<24<27}}} <---> {{{2=8^("1 / 3")<24^("1 / 3")<27^("1 / 3")=3}}}
So {{{2<a[1]<3}}} meaning that {{{2<a[n]<3}}} is true for {{{n=1}}}
Could it be true for all positive integer {{{n}}} values.
If it were true for {{{n=h}}} , we would know that {{{0<a[h]<3}}} .
Then {{{24<a[h]+24<3+24}}}
{{{24<(a[h]+24)<27}}}
{{{24^(1/3)<(a[h]+24)^(1/3)<27^(1/3)=3}}} ,
and since {{{24^("1 / 3")=a[1]}}} and {{{(a[h]+24)^(1/3)=a[h+1]}}} ,
we conclude that {{{2<a[1]<a[h+1]<3}}} .
Since {{{2<a[n]<3}}} is true for {{{n=1}}} ,
and being true for {{{n=h}}} makes it true for {{{n=h+1}}} ,
then {{{2<a[n]<3}}} is true for all values of {{{n}}} .
The integer part of {{{a[n]}}} for any {{{n}}} is {{{2}}} ,
and so the integer part of {{{a[100]}}} is {{{highlight(2)}}} .
It is true that {{{a[n]}}} gets ever closer to {{{3}}} , but it can never get there.