Prove that; Sn = a(r^n - 1)/1

Answer 1

To prove the equation Sn = a(r^n - 1)/1 for a geometric series, we can use the formula for the sum of a geometric series:

Sn = a(1 - r^n) / (1 - r) [Formula for the sum of a geometric series]

Let's simplify this formula step by step:

1. Multiply both sides by (1 - r) to get rid of the fraction:

Sn(1 - r) = a(1 - r^n)

2. Expand the left side using the distributive property:

Sn - Snr = a - ar^n

3. Group the terms containing Sn on the left side:

Sn - a = ar^n - Snr

4. Add a to both sides:

Sn = ar^n - Snr + a

5. Rearrange the terms:

Sn + Snr = ar^n + a

6. Factor out Sn on the left side:

Sn(1 + r) = ar^n + a

7. Divide both sides by (1 + r):

Sn = (ar^n + a) / (1 + r)

8. Simplify the fraction:

Sn = a(r^n + 1) / (1 + r)

Therefore, the correct formula for Sn is Sn = a(r^n + 1) / (1 + r), not Sn = a(r^n - 1) / 1.