Let [tex]h[/tex] be the number of hours that have passed since taking the ibuprofen, and let [tex]A(h)[/tex] be the amount of ibu left after [tex]h[/tex] hours. At the start, all the ibu remains intact, so that [tex]A(0)=400[/tex] mg.
After the first hour, 75% remains, so that [tex]A(1)=A(0)\cdot0.75=300[/tex] mg.
After the second hours, we're left with [tex]A(2)=A(1)\cdot0.75=225[/tex] mg.
And so on. You'll notice that the amount of ibu remaining at hour [tex]h[/tex] forms a geometric sequence with the general form
[tex]A(h)=A(h-1)\cdot0.75[/tex]
We can solve explicitly for [tex]A(h)[/tex]:
[tex]A(1)=A(0)\cdot0.75[/tex]
[tex]A(2)=A(1)\cdot0.75=A(0)\cdot0.75^2[/tex]
[tex]A(3)=A(2)\cdot0.75=A(0)\cdot0.75^3[/tex]
and so on, with the general pattern of
[tex]A(h)=400\cdot0.75^h[/tex]
After 6 hours, we're left with
[tex]A(6)=400\cdot0.75^6\approx71.19[/tex] mg
