\(\frac{30.25.7.8}{75.8.12.14}=\frac{3.10.25.7.8}{25.3.8.12.7.2}=\frac{10}{12.2}=\frac{5.2}{12.2}=\frac{5}{12}\)

a) S=1-3+3²-3³+...+3⁹⁸-3⁹⁹

=>S=(1-3+3²-3³)+(3⁴-3⁵+3⁶-3⁷)+(3⁸-3⁹+3¹⁰-3¹¹)+...+(3⁹⁶-3⁹⁷+3⁹⁸-3⁹⁹)

=>S=-20+3⁴.(1-3+3²-3³)+3⁸.(1-3+3²-3³)+...+3⁹⁶.(1-3+3²-3³)

=>S=-20+3^4.(-20)+3⁸.(-20)+...+3⁹⁶.(-20)

=>S=-20.(1+3⁴+3⁸+...+3⁹⁶)

=>S chia hết cho -20

b) S=S = 1 - 3 + 3² - 3³ + ...  + 3⁹⁸ - 3⁹⁹

=>3S=3-3²+3³-3⁴+...+3⁹⁹-3¹⁰⁰

=>3S+S=(3-3²+3³-3⁴+...+3⁹⁹-3¹⁰⁰)+(1-3+3²-3³+...+3⁹⁸-3⁹⁹)

=>4S=1-3¹⁰⁰

=>S=1-3¹⁰⁰ /4