Candy Pyramid math equation needed!?

Question

So in class we had a pyramid of reeses and we need to find and equation for the number of candies in each row?and an equation for the sum of all the candies? the pyramid has 1 on the first row, 3 on the second, 6 on the 3rd and so on...we found out there is 220 candies in 10 layers. anyone?

Answer 1

The number of items on any layer is figured as:
f(n) = 1 + 2 + 3 + ... + n

These are called the "triangle numbers".
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc.

The short cut for this sum is:
f(n) = n(n+1) / 2
f(1) = 1(2) / 2 = 2/2 = 1
f(2) = 2(3) / 2 = 6/2 = 3
f(3) = 3(4) / 2 = 12/2 = 6
f(4) = 4(5) / 2 = 20/2 = 10
etc.

Then what you want is the sum of these numbers to get the total for "n" layers:
a(n) = f(1) + f(2) + f(3) + ... + f(n)

These are called the "pyramid numbers"
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, etc.

The formula for these numbers, given "n" layers is:

a(n)= (n^3 + 3n² + 2n) / 6

a(1) = (1 + 3 + 2) / 6 = 6/6 = 1
a(2) = (8 + 12 + 4) / 6 = 24/6 = 4
a(3) = (27 + 27 + 6) / 6 = 60/6 = 10
a(4) = (64 + 48 + 8) / 6 = 120/6 = 20
...
a(10) = (1000 + 300 + 20) / 6 = 1320/6 = 220

Answer 2

let us observe the pattern,
1 , 3, 6, ...

we can see that the numbers increase by 1 more than the previous increase. This set of numbers are called "triangular numbers". imagine if you have 1 dot, followed by 2 dots side by side below it, followed by 3 dots below it, it forms a triangle; hence the name.

This can be found by the formula for the nth layer:
n(n+1)/2

*check: 1 => 1*2/2 = 1
2 => 2*3/2 = 3
3 => 3*4/2 = 6

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next is the sum of total for n layers of triangular numbers, these are called "tetrahedral numbers". imagine piling up 1 triangle over another, allowing the each of the vertices of the triangles line up with corresponding vertices on the triangles below it.

The formula for tetrahedral numbers is:
n(n+1)(n+2)/6

*check: 1 => 1*2*3/6 = 1 = 1
2 => 2*3*4/6 = 4 = 1+3
3 => 3*4*5/6 = 10 = 1+3+6
10 => 10*11*12/6 = 220