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Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
A = {5^{50}} - {5^{48}} + {5^{46}} - {5^{44}} + ..... + {5^6} - {5^4} + {5^2} - 1\\
 \Leftrightarrow {5^2}.A = {5^2}.\left( {{5^{50}} - {5^{48}} + {5^{46}} - {5^{44}} + ..... + {5^6} - {5^4} + {5^2} - 1} \right)\\
 \Leftrightarrow 25A = {5^2}{.5^{50}} - {5^2}{.5^{48}} + {5^2}{.5^{46}} - {5^2}{.5^{44}} + .... + {5^2}{.5^6} - {5^2}{.5^4} + {5^2}{.5^2} - {5^2}.1\\
 \Leftrightarrow 25A = {5^{52}} - {5^{50}} + {5^{48}} - {5^{46}} + .... + {5^8} - {5^6} + {5^4} - {5^2}\\
 \Leftrightarrow 25A + A = \left( {{5^{52}} - {5^{50}} + {5^{48}} - {5^{46}} + .... + {5^8} - {5^6} + {5^4} - {5^2}} \right) + \left( {{5^{50}} - {5^{48}} + {5^{46}} - {5^{44}} + ..... + {5^6} - {5^4} + {5^2} - 1} \right)\\
 \Leftrightarrow 26A = {5^{52}} - 1\\
 \Rightarrow A = \dfrac{{{5^{52}} - 1}}{{26}}\\
b,\\
26A + 1 = {5^n}\\
 \Leftrightarrow \left( {{5^{52}} - 1} \right) + 1 = {5^n}\\
 \Leftrightarrow {5^{52}} = {5^n}\\
 \Leftrightarrow n = 52\\
c,\\
26A - 26.\left( {{5^{50}} + 1} \right)\\
 = \left( {{5^{52}} - 1} \right) - \left( {{{26.5}^{50}} + 1} \right)\\
 = {5^{50}}{.5^2} - 1 - {26.5^{50}} - 26\\
 = {25.5^{50}} - {26.5^{50}} - 27\\
 =  - {5^{50}} - 27 < 0\\
 \Rightarrow 26A - 26.\left( {{5^{50}} + 1} \right) < 0\\
 \Leftrightarrow A - \left( {{5^{50}} + 1} \right) < 0\\
 \Leftrightarrow A < {5^{50}} + 1
\end{array}\)