formula for S1 + S2 + S3 + ...+ Sn. Where Sn is the sum of the first n terms of a geometric sequence with the first term of 1 and common ratio of 10 

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.

formula for S1 + S2 + S3 + ...+ Sn. Where Sn is the sum of the first n terms of a geometric sequence with the first term of 1 and common ratio of 10

Explore how to find the sum of the first n terms of a geometric sequence with a starting term of 1 and a common ratio of 10. This solution provides the step-by-step formula and includes examples to illustrate calculations, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation