You are stacking soup cans for a display in a grocery store. Your manager wants you to stack 120 cans in layers, with each layer after the first having one less can then they layer before it. One can should be in the top layer. If you use all 120 cans, how many cans should be in the first layer?
To determine the number of cans in the first layer, we can observe the pattern in the number of cans per layer.
Since each subsequent layer has one less can than the previous layer, we can create a sequence. Let's represent the number of cans in each layer as a series:
First layer: x cans
Second layer: (x - 1) cans
Third layer: (x - 2) cans
...
Last (Nth) layer: (x - (N - 1)) cans
Since the total number of cans is 120, we can express this as an equation:
x + (x - 1) + (x - 2) + ... + (x - (N - 1)) = 120
In this case, we know that the top layer has one can. So the number of layers will be equal to the number of cans in the first layer.
By simplifying the equation, we get:
Nx - (1 + 2 + ... + (N - 1)) = 120
Nx - (N(N-1)/2) = 120
Nx - (N^2 - N)/2 = 120
2Nx - N^2 + N = 240
2Nx = N^2 - N + 240
To find the number of cans in the first layer (x), we can substitute different values for N and solve the equation.
Let's start with N = 2:
2(2)x = 2^2 - 2 + 240
4x = 4 + 238
4x = 242
x = 242/4
x = 60.5
Since we cannot have a fraction of a can, N = 2 is not valid.
Let's try N = 3:
2(3)x = 3^2 - 3 + 240
6x = 9 - 3 + 240
6x = 246
x = 246/6
x = 41
Therefore, there should be 41 cans in the first layer.
To find out how many cans should be in the first layer, we need to determine the pattern of how the number of cans decreases from each layer to the next.
Let's start by listing the number of cans in each layer. In the first layer, there will be N cans. In the second layer, there will be N-1 cans. In the third layer, there will be N-2 cans, and so on.
Now, if we sum up the number of cans in each layer, it should add up to the total number of cans (120 in this case).
The sum of the number of cans in each layer can be calculated using the formula for the sum of an arithmetic series:
Sum = (first term + last term) * number of terms / 2
In our case, the first term is N, and the last term is 1. The number of terms can be found by subtracting 1 from N and adding 1. So, the formula becomes:
120 = (N + 1) * N / 2
To solve this equation, we can multiply both sides by 2, which gives us:
240 = (N + 1) * N
Expanding the right side, we get:
240 = N^2 + N
Rearranging the equation to a quadratic equation form:
N^2 + N - 240 = 0
We can solve this quadratic equation using factoring or the quadratic formula. In this case, factoring would be the most convenient option.
Factoring the quadratic equation, we find:
(N + 16)(N - 15) = 0
This gives us two possible values for N: N = -16 or N = 15.
Since it doesn't make sense to have a negative number of cans in the first layer, we discard N = -16. Therefore, the number of cans in the first layer should be 15.
arithmetic sequence
first term is one so a = 1
d = -1
sum =120
see
http://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html