Formulize this sequence

Question

There is this function defined as;

$$f(x) = 10^x + 10^{x-1} + ...+10^0 $$

Which simply gives the 111.. kind of number, given the length x. What I need to do is a way to formulize this function, find the result value with an algebraic expression. Is this achievable ?

Answer 1

Given function is a geometric summation with first term $1$ and ratio $10$. Thus $$ f(x)=\sum_{n=0}^x 10^n = \frac{10^{x+1}-1}{10-1}=\frac{10^{x+1}-1}{9} $$

Answer 2

Hint: Your series is the sum of the terms of a Geometric progression.

Answer 3

Notice:

• $$\sum_{k=a}^{n}b^{m-k}=\frac{b^{m-n-a}\left(b^{n+1}-b^a\right)}{b-1}$$

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$$f(x)=10^x+10^{x-1}+\dots+10^0=10^{x-0}+10^{x-1}+\dots+10^{x-x}=$$ $$\sum_{k=2}^{x}10^{2-k}=\frac{10-10^{2-x}}{9}$$