Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
*)\\
A = 5 + {5^2} + {5^3} + ..... + {5^{2018}}\\
= \left( {5 + {5^2}} \right) + \left( {{5^3} + {5^4}} \right) + \left( {{5^5} + {5^6}} \right) + ...... + \left( {{5^{2017}} + {5^{2018}}} \right)\\
= \left( {5 + {5^2}} \right) + {5^2}.\left( {5 + {5^2}} \right) + {5^4}.\left( {5 + {5^2}} \right) + .... + {5^{2016}}.\left( {5 + {5^2}} \right)\\
= 30 + {5^2}.30 + {5^4}.30 + ..... + {5^{2016}}.30\\
= 30.\left( {1 + {5^2} + {5^4} + ..... + {5^{2016}}} \right)\,\, \vdots \,\,30\\
*)\\
B = 3 + {3^2} + {3^3} + {3^4} + ..... + {3^{60}}\\
= \left( {3 + {3^2}} \right) + \left( {{3^3} + {3^4}} \right) + ..... + \left( {{3^{59}} + {3^{60}}} \right)\\
= 3.\left( {1 + 3} \right) + {3^3}.\left( {1 + 3} \right) + ..... + {3^{59}}.\left( {1 + 3} \right)\\
= 3.4 + {3^3}.4 + ..... + {3^{58}}.4\\
= 4.\left( {3 + {3^3} + ..... + {3^{58}}} \right)\,\, \vdots \,\,4\\
*)\\
B = 3 + {3^2} + {3^3} + ..... + {3^{60}}\\
= \left( {3 + {3^2} + {3^3}} \right) + \left( {{3^4} + {3^5} + {3^6}} \right) + ...... + \left( {{3^{58}} + {3^{59}} + {3^{60}}} \right)\\
= 3.\left( {1 + 3 + {3^2}} \right) + {3^4}.\left( {1 + 3 + {3^2}} \right) + ..... + {3^{58}}.\left( {1 + 3 + {3^2}} \right)\\
= 3.13 + {3^4}.13 + ..... + {3^{58}}.13\\
= 13.\left( {3 + {3^4} + .... + {3^{58}}} \right)\,\, \vdots \,\,13
\end{array}\)
