Discrete math

Question

The following is a sequence of formula,

$$\begin{align*} \begin{matrix} & 9*1+2 &= &11 \\ & 9*12+3 &= &111 \\ & 9*123+4 &= &1111 \\ & 9*1234+5 &= &11111 \\ \end{matrix} \\ \dots \dots \dots \dots \dots \dots \\ \end{align*}$$

Here numbers are in base $10$.

(a) Establish a formula in $\sum$ notation.

(b) Generalize that formula in for any base $b$..

Comments

for first part -

9 * summation (i=n) to 0 [ ( 10 ) ^n-i * i ]  + (n+1)

Answer 1

(a)- 9* $\sum\limits_{i=0}^{n-1}(n-1)*10^{i}+(n+1)$ where n is the number of digits in your series of 1,12,123 etc

(b)- Little modifications to (a)

Lemme know if I am wrong.

Answer 2

Caption