Giải thích các bước giải:
a, A = 99...9 . \({10^{n + 2}}\) + 25
= (\({10^n}\) - 1).\({10^{n + 2}}\) + 25
= \({10^{2n + 2}}\) - \({10^{n + 2}}\) + 25
= (\({10^{n + 1}}\)² - 2.5.\({10^{n + 1}}\) + 5²
= (\({10^{n + 1}}\) - 5)²
b, B = 99...9 . \({10^{n + 2}}\) + 8.\({10^{n + 1}}\) + 1
= (\({10^n}\) - 1).\({10^{n + 2}}\) + 8.\({10^{n + 1}}\) + 1
= \({10^{2n + 2}}\) - 10.\({10^{n + 1}}\) + 8.\({10^{n + 1}}\) + 1
= \({10^{2n + 2}}\) - 2.\({10^{n + 1}}\) + 1
= (\({10^{n + 1}}\) - 1)²
c, C = 44...4 . \({10^n}\) + 88...8 . 10 + 9
= \(\frac{4}{9}\).(\({10^n}\) - 1).\({10^n}\) + \(\frac{8}{9}\).(\({10^{n - 1}}\) - 1).10 + 9
= \(\frac{4}{9}{10^{2n}} - \frac{4}{9}{10^n} + \frac{8}{9}{10^n} - \frac{{80}}{9} + 9\)
= \(\frac{4}{9}{10^{2n}} + \frac{4}{9}{10^n} + \frac{1}{9}\)
= \({{\rm{[}}\frac{1}{3}({2.10^n} + 1){\rm{]}}^2}\)
d, D = \(\frac{1}{9}.({10^n} - 1){.10^{n + 2}} + \frac{2}{9}.({10^{n + 1}} - 1).10 + 5\)
= \(\frac{1}{9}{.10^{2n + 2}} - \frac{1}{9}{.10^{n + 2}} + \frac{2}{9}{.10^{n + 2}} - \frac{{20}}{9} + 5\)
= \(\frac{1}{9}{.10^{2n + 2}} + \frac{{10}}{9}{.10^{n + 1}} + \frac{{25}}{9}\)
= \({{\rm{[}}\frac{1}{3}({10^{n + 1}} + 5){\rm{]}}^2}\)
