The formula of $(a_1 + a_2 + a_3 + a_4)^2$

Question

I'm trying to calculate $S^2$ with $S = 111\cdots1$ ($n$ number of $1$s) so I'm thinking of rewriting that to $(1 \cdot10^n + 1\cdot10^{n-1} + \cdots)^2$

That's why I want to get the above formula.

Answer 1

Let $111...1$ (n times 1)

$$111...1=10^0+10^1+10^2+...+10^{n-1}=\frac{1-10^n}{1-10}=\frac{10^n-1}{9}$$

$$(10^0+10^1+10^2+...+10^{n-1})^2=(\frac{10^n-1}{9})^2=\frac{10^{2n}-2\times 10^n+1}{81}$$