This image appears to be part of a proof or derivation for the sum of a geometric series. Let’s break it down:

The Statement

The geometric series sum formula is being tested or proven for the series: $S_n = a + ar + ar^2 + \dots + ar^{n-1},$ and its closed form: $S_n = a \frac{1 - r^n}{1 - r}, \quad r \neq 1.$

Step in the Proof

• For $n = 1$:
  • Left-Hand Side (LHS): $S_1 = a,$ since only the first term is included.
  • Right-Hand Side (RHS): Substitute $n = 1$ into the formula: $S_1 = a \frac{1 - r^1}{1 - r}.$ Simplify the numerator: $S_1 = a \frac{1 - r}{1 - r} = a,$ which matches the LHS.

This confirms the formula works for $n = 1$. The next steps would likely involve proving the formula for $n > 1$, possibly using mathematical induction.

Would you like to continue with the full proof using induction, or clarify any steps?

Related Questions

1. How does the formula for a finite geometric series work for $r > 1$?
2. What happens when $r = 1$ in the series formula?
3. Can we derive this formula from first principles without induction?
4. How is the infinite geometric series related to this formula?
5. What real-world problems can be modeled with a geometric series?

Tip

When solving series problems, always verify boundary cases like $n = 1$ or $r = 1$ for correctness!