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How can I calculate the following number: $$ \sqrt{444 \cdots (2n \text{ digits}) + 111 \cdots (n+1 \text{ digits}) - 666 \cdots (n \text{ digits})}.$$

My trying :

I have tried to calculate these data by observing the pattern of $ 7^2 , 67 ^2 ,667^2 , 6667^2 , \dots$. But there is no promising news to solve it by this process.

Way to infinity
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  • "calculate this data" ought to be "calculate these data" – Trancot Nov 29 '13 at 10:24
  • Could you be more clear what your "calculation" is? – Trancot Nov 29 '13 at 10:25
  • Related : http://math.stackexchange.com/questions/581642/333-dots33-ldots3-10n1-9n-10-over-27 , http://math.stackexchange.com/questions/360137/recursive-sequence-with-elusive-formula and http://math.stackexchange.com/questions/557977/what-is-the-concept-behind-divisibility-of-large-numbers-that-contain-only-the-d – lab bhattacharjee Nov 29 '13 at 12:34

1 Answers1

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Hint:

Use the fact that $111\dots 111 = \frac{10^n-1}{9}$ (where there are $n$ ones)

Then your number is

$$4\frac{10^{2n}-1}{9} + \frac{10^{n+1}-1}{9} - 6\frac{10^n-1}{9} = \frac{4\times 10^{2n}-4 +10^{n+1}-1-6\times 10^n+6}{9}$$

$$=\frac{4\times 10^{2n}+4\times 10^n+1}{9}$$

$$= \frac{(2\times 10^n+1)^2}{9}$$

Old John
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