Question

REPEDE DAU COROANA !!​

Answer 1

a)

$\displaystyle \frac{7^{n} +7^{n+1} +7^{n+2}}{2^{n} +2^{n+1} +2^{n+4}} =\frac{7^{n} (1+7+7^{2})}{2^{n}(1+ 2 +2^{4})} =\frac{7^{n}\cdot57}{2^{n} \cdot19} =\frac{3\cdot7^{n}}{2^{n}}$

b)

$\displaystyle \frac{2^{2n} +2^{2n+1} +2^{n+2}+2^{2n+3}}{4^{n} +4^{n+1} +4^{n+2}+4^{n+3}} =\frac{2^{2n} (1+2+2^{2}+2^{3} )}{4^{n}(1+ 4 +4^{2}+4^{3}) } =\frac{2^{2n}\cdot15}{(2^{2} )^{n} \cdot85} =\frac{15}{85}=\frac{3}{17}$

c)

$\displaystyle \frac{3^{n+2}\cdot5^{n} +3^{n} \cdot5^{n+1} }{3^{n+1}\cdot5^{n} +2\cdot3^{n+1} \cdot5^{n}} =\frac{3^{n}\cdot5^{n}(3^{2}+5) }{3^{n+1}\cdot5^{n}(1+ 2)} =\frac{14}{3\cdot3} =\frac{14}{9}$

d)

$\displaystyle \frac{2^{n}\cdot3^{n+1} +2^{n+2} \cdot3^{n} }{3^{n}\cdot5^{n+1} +3^{n+2} \cdot5^{n}} =\frac{2^{n}\cdot3^{n}(3 +2^{2}) }{3^{n}\cdot5^{n}(5 +3^{2})} =\frac{2^{n}\cdot7 }{5^{n}\cdot14}=\frac{2^{n} }{5^{n}\cdot2}=\frac{2^{n-1} }{5^{n}}$