Ashop owner has 500 cans of tomatoes that are near their expiration date. she decides to display them in a triangular pyramid and lower the price to try and sell them quickly. how many cans will be on the bottom layer, and will there be any cans left over?

How many cans will be on the bottom layer, and will there be any cans left over?

On the bottom layer will be 31 cans  and there will be 4 cans left over

Step-by-step explanation:

To resolve this problem we need to know Gauss sum of natural numbers, from k = 1 to n:

∑(1)

We want to know the value of n when the sum is equal or close to 500, so we replace 500 in the equation (1) to find n:

(2)

We need to use the quadratic formula to resolve the equation (2)

(3)

The coefficients according to equation (2) are:

a=1

b=1

c= 1000

Now, using the quadratic formula we can find n :

Positive sign before the square root,

Negative sign before the square root,

The negative term doesn't have sense in this case, We will use the positive term, the whole number closest to the answer we got when clearing n is 31, we will replace n=31 in the Gauss sum of natural numbers equation. (1)

∑

∑

∑

∑

∑

Now, we know the triangular pyramid will be 31 cans on the bottom layer and there will be 4 cans left over (500-496=4)