So there is this question ( question1) " 1. Company Zs dividends per share are expected to grow indefinitely by 5% a year. If next years dividend is $50 and the market capitalization rate (the return required by shareholders) is 10%, what is the current stock price?"

its answer is Q2 P = D/(rg) = 50 / (10%5%) = 1000

Then there is this question (question 2, a,b,c)

"a) Company Y is expected to pay an end-of-year dividend of $10 a share. After the dividend its stock is expected to sell at $200. If the market capitalization rate is 12%, what is the current stock price? b") Company Z does not plow back any earnings and is expected to produce a level dividend stream of $5 a share. If the current stock price is $60, what is the market capitalization rate? c)Company Ms dividends per share are expected to grow indefinitely by 5% a year. If it just paid a dividend (D1)of $10 and the market capitalization rate is 10%, what is the current stock price(P0)?

and its answers are "

a. P0 = (P1+Div1) / (1+r) = (200+10)/1.12 = $187.50

b. P0 = Div1 / r , then, r = Div1 / P0 = 5/60 = 8.33%

c. P0 = Div1 / (r-g), Div0 = 10 Div1 = Div0*(1+g) = 10*1.05 = 10.5, then, P0 =10.5/(0.1-.05) = $210

"

So my questions are , what formulas were used or derived from to solve for all these questions ( question1, and question2) and why is the value of the dividend in question 2 part c ,10.5, rather than just 10? This makes no sense to me since for question one we keep the value 50, why are we not making it 52.5 and just using 50??