\begin{array}{l} 1)\\ a)B = 1 + 5 + {5^2} + {5^3} + ... + {5^{100}}\\ \Rightarrow 5.B = 5 + {5^2} + {5^3} + ... + {5^{101}}\\ \Rightarrow 5B - B = 4B = {5^{101}} - 1\\ \Rightarrow B = \dfrac{{{5^{101}} - 1}}{4}\\ b)D = 1 + 9 + {9^2} + ... + {9^{2019}}\\ \Rightarrow 9D = 9 + {9^2} + {9^3} + .. + {9^{2020}}\\ \Rightarrow 8D = {9^{2020}} - 1\\ \Rightarrow D = \dfrac{{{9^{2020}} - 1}}{8}\\ B2)\\ a)A = 2 + {2^2} + {2^3} + ... + {2^{60}}\\ = \left( {2 + {2^2}} \right) + \left( {{2^3} + {2^4}} \right) + ... + \left( {{2^{59}} + {2^{60}}} \right)\\ = 2\left( {1 + 2} \right) + {2^3}\left( {1 + 2} \right) + ... + {2^{59}}\left( {1 + 2} \right)\\ = 2.3 + {2^3}.3 + ... + {2^{59}}.3\\ = \left( {2 + {2^3} + ... + {2^{59}}} \right).3 \vdots 3\\ A = \left( {2 + {2^3}} \right) + \left( {{2^2} + {2^4}} \right) + ... + \left( {{2^{58}} + {2^{60}}} \right)\\ = 2\left( {1 + {2^2}} \right) + {2^2}.\left( {1 + {2^2}} \right) + ... + {2^{58}}\left( {1 + {2^2}} \right)\\ = 2.5 + {2^2}.5 + ... + {2^{58}}.5\\ = \left( {2 + {2^2} + {2^5} + ... + {2^{58}}} \right).5 \vdots 5\\ A = \left( {2 + {2^2} + {2^3}} \right) + ... + \left( {{2^{58}} + {2^{59}} + {2^{60}}} \right)\\ = 2\left( {1 + 2 + {2^2}} \right) + .. + {2^{58}}\left( {1 + 2 + {2^2}} \right)\\ = 2.7 + {2^4}.7 + ... + {2^{58}}.7\\ = \left( {2 + {2^4} + ... + {2^{58}}} \right).7 \vdots 7\\ b)B = 7 + {7^2} + {7^3} + ... + {7^{99}} + {7^{100}}\\ = 7\left( {1 + 7} \right) + {7^3}.\left( {1 + 7} \right) + ... + {7^{99}}\left( {1 + 7} \right)\\ = \left( {7 + {7^3} + ... + {7^{99}}} \right).8 \vdots 8\\ B = \left( {7 + {7^3}} \right) + \left( {{7^2} + {7^4}} \right) + ... + \left( {{7^{98}} + {7^{100}}} \right)\\ = 7\left( {1 + {7^2}} \right) + {7^2}\left( {1 + {7^2}} \right) + ... + {7^{98}}\left( {1 + {7^2}} \right)\\ = \left( {7 + {7^2} + {7^5} + {7^6} + ... + {7^{98}}} \right).50 \vdots 50 \end{array}
