Ta có:11...1(n chữ số)=\(\left(10^n-1\right):9\)
\(\Rightarrow A=2\left(1+11+...+11...1\right)\)
\(=2\left(10-1+10^2-1+...+10^{50}-1\right):9\)
\(=2\left(10+10^2+...+10^{50}-50\right):9\)
\(\Rightarrow\frac{2A}{9}=10\left(1+10+10^2+...+10^{49}\right)-50\)
\(=10\left(10^{50}-1\right):9-50=10\left(10^{50}-5-1\right):9\)
\(\Leftrightarrow A=\frac{20\left(10^{50}-6\right)}{81}\)
Đặt A = 2 + 22 + 222 +....+ 22..2 (50 cs2)
= 2 ( 1 + 11 + 111 + ... + 11..1( 50 cs 1)
9A = 2( 9 + 99 + 999 +...+ 99..9 ( 50 cs9)
= 2[(10 + 100 + 1000 +...+ 1 00..0( 50 cs0) - 50]
= 2[(101 + 102 +....+ 1050) - 50]
....Trong () dễ r tự tính
=> 9A = 2[ \(\frac{10^{51}-10}{9}-50\)]
\(9A=2.\frac{10^{51}-460}{9}\)
\(A=\frac{2\left(10^{51}-460\right)}{81}\)
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