Giải thích các bước giải:
\(\begin{array}{l}
3,\\
I = 5 + {5^2} + {5^3} + ..... + {5^{2016}}\\
\Leftrightarrow 5I = 5.\left( {5 + {5^2} + {5^3} + ..... + {5^{2016}}} \right)\\
\Leftrightarrow 5I = {5^2} + {5^3} + {5^4} + .... + {5^{2017}}\\
\Rightarrow 5I - I = \left( {{5^2} + {5^3} + {5^4} + .... + {5^{2017}}} \right) - \left( {5 + {5^2} + {5^3} + ..... + {5^{2016}}} \right)\\
\Leftrightarrow 4I = {5^{2017}} - 5\\
4I + 5 = {5^x}\\
\Leftrightarrow \left( {{5^{2017}} - 5} \right) + 5 = {5^x}\\
\Leftrightarrow {5^{2017}} = {5^x}\\
\Leftrightarrow x = 2017\\
4,\\
E = 1 + 3 + {3^2} + {3^3} + .... + {3^{98}}\\
= {3^0} + {3^1} + {3^2} + {3^3} + .... + {3^{98}}\\
= \left( {{3^0} + {3^1} + {3^2}} \right) + \left( {{3^3} + {3^4} + {3^5}} \right) + ...... + \left( {{3^{96}} + {3^{97}} + {3^{98}}} \right)\\
= \left( {1 + 3 + {3^2}} \right) + {3^3}.\left( {1 + 3 + {3^2}} \right) + .... + {3^{96}}.\left( {1 + 3 + {3^2}} \right)\\
= 13 + {13.3^3} + .... + {13.3^{96}}\\
= 13.\left( {1 + {3^3} + .... + {3^{96}}} \right)\,\, \vdots \,\,13\\
5,\\
{3^{500}} = {\left( {{3^5}} \right)^{100}} = {243^{100}}\\
{7^{300}} = {\left( {{7^3}} \right)^{100}} = {343^{100}}\\
243 < 343 \Rightarrow {243^{100}} < {343^{100}}\\
\Rightarrow {3^{500}} < {7^{300}}
\end{array}\)
