The sum $S_n$ of the first $n$ terms of a geometric sequence with the first term $a = 1$ and common ratio $r = 10$ is given by:

$S_n = a \frac{r^n - 1}{r - 1}$

Since $a = 1$ and $r = 10$, substitute these values into the formula:

$S_n = 1 \cdot \frac{10^n - 1}{10 - 1}$

Simplifying further:

$S_n = \frac{10^n - 1}{9}$

Thus, the formula for the sum $S_n$ of the first $n$ terms is:

$S_n = \frac{10^n - 1}{9}$

Example

For $n = 3$: $S_3 = \frac{10^3 - 1}{9} = \frac{1000 - 1}{9} = \frac{999}{9} = 111$

Would you like more details or have any questions on this formula?

Related Questions

1. How can we derive the sum formula for a geometric sequence in general?
2. What happens to $S_n$ as $n \to \infty$ when $r > 1$?
3. Can you show the sum of a geometric sequence with a different common ratio, say $r = 2$?
4. How would the sum formula change if the first term $a$ were a different number?
5. How is the sum formula different for an infinite geometric series?

Tip

For quick mental math with geometric sequences, recognize patterns in powers of the common ratio to approximate sums.