<s>[INST] Prove that every number of the sequence $49,4489,444889,\ldots$ is a perfect square 
Prove that every number of the sequence $49,4489,444889,\ldots$ is a perfect square. 

We can observe that $49=7^2, 4489=67^2, 444889=667^2, \ldots$
I have tried expanding terms of the sequence, and to express it as a whole square. But it was too tough for me. 
Any help will be appreciated. 
 [/INST] We can prove it as follows:
\begin{align*}
\underbrace{666\ldots6}_{n}\,7^2
&= \left[7 + 6(10^1 + 10^2 + \cdots + 10^n) \right]^2 \\
&= \left[7 + 6 \frac{10^{n+1} - 10}{9}\right]^2 \\
&= \left[ \frac{2 \cdot 10^{n+1} + 1}{3}\right]^2 \\
&= \frac{4 \cdot 10^{2n+2} + 4 \cdot 10^{n+1} + 1}{9} \\
&= 4 \cdot \frac{10^{2n+2} - 1}{9} + 4 \cdot \frac{10^{n+1} - 1}{9} + 1 \\
&= 4 \cdot \underbrace{111 \ldots 1}_{2n+2} + 4 \cdot \underbrace{111 \ldots 1}_{n+1} + 1 \\
&= \underbrace{444 \ldots 4}_{2n+2} + \underbrace{444 \ldots 4}_{n+1} + 1 \\
&= \underbrace{444 \ldots 4}_{n+1}\,\underbrace{888 \ldots 8}_{n}\,9.
\end{align*}
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