So we have 1280,640,320,...
This is a geometric sequence with the first term, [tex] a_{1} =1280[/tex]. To find the common ratio r, we are going to divide any current term by a previous one: [tex]r= \frac{640}{1280} =(0.5)[/tex]
Remember that the main formula of a geometric sequence is:
[tex] a_{n} = a_{1} r^{n-1} [/tex]
Where [tex] a_{n} [/tex] is the nth term (in our case 40), [tex] a_{1} [/tex] is the first term (in our case 1280), [tex]r[/tex] is the common ratio (0.5), and [tex]n[/tex] is the position of the term in the sequence (in our case our weeks)
Now we can replace the values to get:
[tex]40=1280(0.5)^{n-1} [/tex]
[tex](0.5)^{n-1} = \frac{40}{1280} [/tex]
[tex](0.5)^{n-1} =0.03125[/tex]
Since our variable, n, is the exponent, we are going to use logarithms to bring it down:
[tex]ln(0.5)^{n-1} =ln(0.03125)[/tex]
[tex](n-1)ln(0.5)=ln(0.03125)[/tex]
The only thing left now is solving for n to find our week:
[tex]n-1= \frac{ln(0.03125)}{ln(0.5)} [/tex]
[tex]n-1=5[/tex]
[tex]n=6[/tex]
We can conclude that in the sixth week the cafeteria will sell 40 slices of pizza.
