Question

A school cafeteria sold 1,280 slices of pizza the first week,640 the second week,and 320 the third week.if this pattern continues,in what week will the cafeteria sell 40 slices?

Answer 1

So we have 1280,640,320,... 
This is a geometric sequence with the first term, $a_{1} =1280$. To find the common ratio r, we are going to divide any current term by a previous one: $r= \frac{640}{1280} =(0.5)$

Remember that the main formula of a geometric sequence is:
$a_{n} = a_{1} r^{n-1}$
Where $a_{n}$ is the nth term (in our case 40), $a_{1}$ is the first term (in our case 1280), $r$ is the common ratio (0.5), and $n$ is the position of the term in the sequence (in our case our weeks)

Now we can replace the values to get:
$40=1280(0.5)^{n-1}$
$(0.5)^{n-1} = \frac{40}{1280}$
$(0.5)^{n-1} =0.03125$
Since our variable, n, is the exponent, we are going to use logarithms to bring it down:
$ln(0.5)^{n-1} =ln(0.03125)$
$(n-1)ln(0.5)=ln(0.03125)$
The only thing left now is solving for n to find our week:
$n-1= \frac{ln(0.03125)}{ln(0.5)}$
$n-1=5$
$n=6$

We can conclude that in the sixth week the cafeteria will sell 40 slices of pizza.