How do you find equation for a pyramid of oranges stacked 1 2x2 3x3 4x4?

Answer 1

The equation for n layers is S(n) = n(n+1)(2n+1)/6

It is simplest to prove it by induction.

When n = 1,

S(1) = 1*(1+1)(2*1+1)/6 = 1*2*3/6 = 1.

Thus the formula is true for n = 1.

Suppose it is true for n = m. That is, for a pyramid of m levels,

S(m) = m*(m+1)*(2m+1)/6

Then the (m+1)th level has (m+1)*(m+1) Oranges and so

S(m+1) = S(m) + (m+1)*(m+1)

= m*(m+1)*(2m+1)/6 + (m+1)*(m+1)

= (m+1)/6*[m*(2m+1) + 6(m+1)]

= (m+1)/6*[2m^2 + m + 6m + 6]

= (m+1)/6*[2m^2 + 7m + 6]

= (m+1)/6*(m+2)*(2m+3)

= (m+1)*(m+2)*(2m+3)/6

= [(m+1)]*[(m+1)+1)]*[2*(m+1)+1]/6

Thus, if the formula is true for n = m, then it is true for n = m+1.

Therefore, since it is true for n =1 it is true for all positive integers.