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August 2022 1 76 Report

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nafanya2014

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А)
$\frac{2\cdot 3^{n+2}-5\cdot 3^{n+1}}{3^{n-1}}= \frac{ 3^{n+1}\cdot (2\cdot 3^{n+2-(n+1)}-5)}{3^{n-1}}= \frac{ 3^{n+1}\cdot (2\cdot 3-5)}{3^{n-1}}=\frac{ 3^{n+1}}{3^{n-1}}= \\ \\ =3^{(n+1)-(n-1)}=3^2=9$

б)
$\frac{25\cdot 4^n}{4^n-4^{n-1}}= \frac{25\cdot 4^n}{4^{n-1}\cdot (4-1)}}= \frac{25\cdot 4}{ (4-1)}}= \frac{100}{3} =33 \frac{1}{3}$

в)
$\frac{10\cdot 6^{n}}{2^{n+1}\cdot 3^{n-1}} =\frac{10\cdot 6^{n}}{2^{n}\cdot 2\cdot 3^{n}\cdot 3^{-1}} =\frac{10\cdot 6^{n}}{6^{n}\cdot 2\cdot 3^{-1}} =15$

г)
$\frac{2^{2n-1}\cdot 5^{2n+1}}{100^n}= \frac{2^{2n}\cdot 2^{-1}\cdot 5^{2n}\cdot 5}{100^n}= \frac{10^{2n}\cdot 2^{-1}\cdot 5}{100^n}= \frac{100^{n}\cdot 5}{2\cdot 100^n}= \frac{5}{2}=2,5$

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