Question 4
A sequence of numbers is added in the following way:
For example, for the sequence $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\ldots\}$, all the numbers in the sequence are first written as decimals as shown below and then added up together.
$$\begin{array}{r}0.1\\0.02\\0.003\\0.0004\\0.00005\\0.000006\\0.0000007\\0.00000008\\0.000000009\\0.0000000010\\0.00000000011\\0.000000000012\\+\quad\vdots\\\hline0.123456790123\ldots\end{array}$$
What is the result when the sequence $\{9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117,\ldots\}$ is added up in the same way?
For example, for the sequence $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\ldots\}$, all the numbers in the sequence are first written as decimals as shown below and then added up together.
$$\begin{array}{r}0.1\\0.02\\0.003\\0.0004\\0.00005\\0.000006\\0.0000007\\0.00000008\\0.000000009\\0.0000000010\\0.00000000011\\0.000000000012\\+\quad\vdots\\\hline0.123456790123\ldots\end{array}$$
What is the result when the sequence $\{9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117,\ldots\}$ is added up in the same way?