Question

To determine ${\mathbf{\text{R}}}_{\mathbf{\text{n}}}$using iteration.

Short answer

$R_n=2-n+n^2$

Step-by-step solution

Given data

No three of the great circles go through the same point.

Concept used of recurrence relation

A recurrence relation is an equation that recursively defines a sequence where the next term is a function of the previous terms (Expressing ${\mathit{F}}_{\mathbf{n}}$ as some combination of ${\mathit{F}}_{\mathbf{i}}$with $\mathit{i}\mathbf{<}\mathit{n}\mathbf{)}$.

Solve by recurrence relation

Let us use the recurrence relation \({R_0} = 0,{R_n} = {R_{n - 1}} + 2(n - 1)\).\(\begin{aligned}{}{R_n} &= {R_{n - 1}} + 2(n - 1)\\ &= {R_{n - 2}} + 2(n - 1) + 2(n - 1)\\ &= {R_{n - 3}} + 2(n - 3) + 2(n - 1) + 2(n - 1)\\ &= {R_1} + 2 + \ldots + 2(n - 3) + 2(n - 2) + 2(n - 1)\\ &= 2 + 2 + \ldots + 2(n - 3) + 2(n - 2) + 2(n - 1)\\ &= 2 + 2 + \ldots + 2(n - 3) + 2(n - 2) + 2(n - 1)\\ &= 2 + \sum\limits_{k = 0}^{n - 1} 2 k\\ &= 2 + 2 \cdot \frac{{n(n - 1)}}{2}\\ &= 2 - n + {n^2}\end{aligned}\)

Thus, \({R_n} = 2 - n + {n^2}\).