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= (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) = (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= (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.