Display of cups in triangle/pyramid

Question

In this above picture, there are 15 cups, to make a perfect pyramid, you start the bottom row with 5 cups.

What is the formulae to start the bottom row, if I have $n$ cups.

Answer 1

Let's start with the other way round. If the base has $k$ cups, the total number of cups $n$ will be given by:

$$n=k+(k-1)+(k-2)+\ldots+1=\frac{k^2+k}{2}$$ Or: $$k^2+k-2n=0$$ Now, just solve the quadratic equation to get: $$k=\frac{-1\pm\sqrt{1+8n}}{2}$$ Only one of these solution will be positive, and of course, only certain $n$ will have integer $k$.

Answer 2

first you must find the explicit formula that gives you the number of cups in a row by plugging in the row number... first we find our difference since this pyramid starts with 5 then goes to 4,3,2,1... we know our difference or d=-1 and that row 1 or r[1] = 5
r[0]= r[1]-d = 5--1 = 5+1 = 6

r[n]= d(n)+r[0]

r[n]= -1(n)+6 <--Explicit Formula

now we put the formula in summation form: r= number of rows $$\sum_{n=1}^r = -1(n)+6 $$

plug into calculator and it should give you the number of cups.

So... $$\sum_{n=1}^5 = -1(n)+6 = 15 $$

Use this equation in the calculator $$\sum_{x=1}^5 = -1(x)+6 = 15 $$