bus holds 66 people. Starts out empty. Picks up 1 person at the 1st station. 2nd person at the 2nd station 3rd person at the 3rd station and so on. No one gets off. After how many stations will fill up the bus Show your work- explain your thinking?

Wouldn't the last person get on at the 66th station?

the way you worded the question it simply becomes

1+1+1+1+... = 66
clearly 66 ones are needed, so after 66 stations the bus is full

if you meant 1st station one person gets one, on the 2nd station two people get on, on the third 3 people get on, etc
then it becomes a sequence/series type of question and you would have

1+2+3+4+... = 66
here a=1
d=1 and n is unknown but S(n) = 66
S(n) = n/2[2a + d(n-1)]
66 = n/2[2 + 1(n-1)]
132 = n[1+n]
n^2 + n - 132 = 0
(n+12)(n-11) = 0
n = -12 or n = 11
we need the positive integer solution which is 11
test: S(11) = 11/2[2 + 10] = 66

To find out how many stations it will take for the bus to be completely filled, we can start by understanding the pattern of passengers boarding the bus.

The number of passengers boarding the bus follows the pattern of adding one additional person at each station.

At the first station, the bus picks up 1 person. At the second station, it picks up 2 people (1 from the first station and 1 from the second station), at the third station, it picks up 3 people (1 from the first, 1 from the second, and 1 from the third), and so on.

We can observe that the number of people picked up at each station follows the sequence of natural numbers: 1, 2, 3, 4, 5, and so on.

To find the number of stations it will take to fill the bus, we need to find the value of n, where the sum of the first n natural numbers is equal to or greater than the bus capacity of 66.

The sum of the first n natural numbers can be calculated using the formula: S = (n/2) * (first term + last term), where S is the sum and n is the number of terms.

Using this formula, we need to find the value of n for which the sum of the first n natural numbers is equal to or greater than 66.

Using trial and error, we can find that the first 11 natural numbers have a sum of 66: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66.

Therefore, it will take 11 stations for the bus to be completely filled.

So, after 11 stations, the bus will be full, with a total of 66 people on board.

Note: To verify this answer, you can also calculate the sum of the first n natural numbers until the sum exceeds 66.