4. Express the sum $8+88+888+8888+\ldots+8 \ldots 8$ in terms of n, if the last term in its notation contains n eights
# Solution

$8+88+888+8888+\ldots+8 \ldots 8=8(1+11+111+\ldots+1 \ldots 1)$

Notice that

$$
\begin{gathered}
10-1=9 * 1 \\
100-1=9 * 11 \\
\ldots \ldots \ldots \ldots \\
10^{n}-1=9 * 1 \ldots 1
\end{gathered}
$$

Then $\frac{8}{9}\left(\frac{10 \times 10^{n}-10}{9}-n\right)=\frac{8}{81}\left(10^{n+1}-10-9 n\right)$.

## Grading Criteria

Only the correct answer - 0 points.

A method for calculating the sum is proposed and some calculations are performed - 1-2 points depending on progress in the problem.

A method for calculating the sum is indicated, but errors are made in formulating expressions 3-5 points.

The correct answer is obtained justifiably - 7 points.