Let P_n denote the total number of triangles formed by joining the vertices of a regular polygon having n sides.

If:

P_n+1 − P_n = 66

Find n.

Number of triangles formed from n vertices:

P_n = ⁿC₃

Therefore:

P_n+1 − P_n = ⁿC₂

Given:

ⁿC₂ = 66

n(n−1)/2 = 66

n(n−1) = 132

n²−n−132 = 0

(n−12)(n+11)=0

n = 12

Final Answer

12