How to derive geometric sequence formula?

The referenced YouTube video discusses how to derive the sum formula of a finite geometric series, not the formula for a geometric sequence itself. Therefore, I'll explain both how to find a general term in geometric sequence and how to derive the formula for the sum of a finite geometric series.

Geometric Sequence: Finding a General Term

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant called the common ratio, r.

• First Term: a₁
• Second Term: a₂ = a₁ * r
• Third Term: a₃ = a₂ r = a₁ r r = a₁ r²
• Fourth Term: a₄ = a₃ r = a₁ r² r = a₁ r³

Following this pattern, the nth term (a_n) of a geometric sequence can be expressed as:

*a_n = a₁ r^(n-1)**

This is the general formula for the nth term of a geometric sequence, where:

• a_n is the nth term
• a₁ is the first term
• r is the common ratio
• n is the term number

Deriving the Sum Formula of a Finite Geometric Series

A geometric series is the sum of the terms in a geometric sequence. A finite geometric series is the sum of a limited number of terms. Let S_n represent the sum of the first n terms of a geometric series:

S_n = a₁ + a₁r + a₁r² + ... + a₁r^(n-1)

To derive the formula for S_n, we can follow these steps as described in the reference video:

1. Multiply the series by r:

   rS_n = a₁r + a₁r² + a₁r³ + ... + a₁rⁿ

2. Subtract the second equation from the first:

   S_n - rS_n = (a₁ + a₁r + a₁r² + ... + a₁r^(n-1)) - (a₁r + a₁r² + a₁r³ + ... + a₁rⁿ)

   Notice that most terms cancel out, leaving:

   S_n - rS_n = a₁ - a₁rⁿ

3. Factor out S_n on the left and a₁ on the right:

   S_n(1 - r) = a₁(1 - rⁿ)

4. Solve for S_n:

   S_n = a₁(1 - rⁿ) / (1 - r)

This formula calculates the sum of the first n terms of a geometric series, where:

• S_n is the sum of the first n terms
• a₁ is the first term
• r is the common ratio
• n is the number of terms

The video confirms this derivation, stating that the sum of a finite geometric series is a₁(1 - rⁿ) / (1 - r).