2 Spheres can be stacked to build a square pyramid as shown

(2) Spheres can be stacked to build a square pyramid as shown below: Aflat Top View 0 HRMS 14 Spheres Front Vievw Lamsim Enterprises Inc. 2015 (A) top and side view of stacked spheres (B) stacked cannonballs The number of spheres, S(n), needed to build a square pyramid with n layers is given by 3 S(n) =-+-+ n For example in figure A, the pvramid shown is made up of of 3 lavers of spheres Therefore there are S(3) 33/3432/2+ 3/6-14 spheres in this pyramid. Use the function S(n) to answer the following questions

Solution

number of spheres is given by S(n) = n³/3+n²/2+n/6

for 3 layers S(n) = 14= 1+4+9

for 4 layers S(n) = 30 = 1+4+9+16

we can see the pattern is the sum of squares i.e for n layers sum of square uptill n.

so function which defines number of spheres in the bottom layer of the pyramid of n layers= n²