Question

Calculati suma: 9+99+999+...+99...9
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De 1998 ori 9

Answer 1

[tex]A=9+99+999+...+999...9999\\ A=9(1+11+111+....+111...11)\\ A=9( \frac{10-1}{10-1}+ \frac{10^2-1}{10-1}+...+ \frac{10^{1999}-1}{10-1})\\ A=9* \frac{1}{10-1}(10-1+10^2-1+...+10^{1999}-1)\\ A=\not9* \frac{1}{\not9}( \frac{10^{2000}-10}{9}-1999)\\ A= \frac{10^{2000}-10-17991}{9}\\ A= \frac{10^{2000}-18001}{9} [/tex]