### Instruction: The number of cans in the layers of a display in a supermarket form an arithmetic sequence. The bottom layer has 28 cans; the next layer has 25 cans and so on until there is one can at the top of the display. How many cans are in the entire display?### Response: To find the total number of cans in the display, I need to add up all the terms of the arithmetic sequence.
An arithmetic sequence is a sequence where each term is obtained by adding or subtracting a constant value from the previous term.
In this case, the constant value is -3, since each layer has 3 cans less than the layer below it.
To add up an arithmetic series, I can use the formula S_n = n/2 (a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
I need to find out how many terms (or layers) there are in the sequence, so I can plug in n.
To do that, I can use the formula a_n = a_1 + (n-1) d, where d is the common difference.
I know that a_1 is 28, a_n is 1, and d is -3, so I can plug them in and get 1 = 28 + (n-1) (-3).
Simplifying this equation, I get -27 = (n-1) (-3), or 9 = n-1, or n = 10.
So there are 10 layers in the display, and 10 terms in the sequence.
Now I can plug in n = 10, a_1 = 28, and a_n = 1 into the formula for S_n and get S_10 = 10/2 (28 + 1).
Simplifying this, I get S_10 = 5 (29), or S_10 = 145.
Therefore, the total number of cans in the display is 145.