Angles' ratio

I found 2 ways to solve this problem.

Way 1:

$n$ = number of sides

$α$ = interior angle

$n \times (π-α) = 2π$

$α = π-$ $\frac{2π}{n}$ $= π$ ( $1-$ $\frac{2}{n}$ ) $= π$ ( $\frac{n-2}{n}$ ) $=$ $\frac{π}{n}$ $(n-2)$

$\frac{α_{102}}{α_{101}}$ = $\frac{(102-2)/102}{(101-2)/101}$ = $\frac{50}{51}$ $\times$ $\frac{101}{99}$ = $\frac{5050}{\boxed{5049}}$

Way 2:

There is actually a pattern!

Formulas:

First polygon's number of sides = $f$

Term $y$ = $\frac{(f-1) \times (f)}{2}$

{**Note that this is also the formula for triangular number!! where n = (f-1); refer to https://en.wikipedia.org/wiki/Triangular_number }

Term $x$ = Term $y$ - 1

Use them:

Term $y$ = $\frac{100 \times 101}{2}$ = $\frac{10100}{2}$ = $5050$

Term $x$ = $5050-1$ = $\boxed{5049}$