I have a number theory problem with solution but i dont understand why one can write this equallity (green question mark). Because 1122 is not 11*22 or 111222 is not 111*222.
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I have a number theory problem with solution but i dont understand why one can write this equallity (green question mark). Because 1122 is not 11*22 or 111222 is not 111*222.
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The attachment has a typo. It should be:
$\displaystyle 11..122...2 = \frac 1 9 (10^{2n} - 10^n) + \frac 2 9 (10^n - 1) $
The rest is correct. For example, starting with 1122 (n=2) you have:
$\displaystyle 1122 = 1100 + 22 = \frac 1 9 9900 + \frac 2 9 99 = \frac 1 9 (10^4 - 10^2) + \frac 2 9 (10^2 -1) $
$\displaystyle = \frac 1 9 (10^4 +10^2 - 2) = \frac 1 9 (10^2-1)(10^2+2) = \frac 1 9 99 \cdot 102 = 33 \cdot 34$
To me it looks like the expression immediately after the green question mark and before the next equal sign...
1/9(10^n -1).2/9(10^n -1)
i think that dot separates 1's and 2's
Examples: n = 1
1/9(9).2/9(9) = 1.2, '1 dot 2' or twelve
n = 2
1/9(99).2/9(99) = 11.22, '11 dot 22' or 1122
n = 3
1/9(999).2/9(999) = 111.222, '111 dot 222' or 111222
Am I stretching it too much?