How to Write a Recursive Formula for a Geometric Sequence | Khan Academy

Identifying a Geometric Sequence

• 🎯 The video demonstrates how to find the recursive formula for a geometric sequence using the first four terms provided.
• 🔍 To determine if the sequence is geometric, we check if there's a common ratio between consecutive terms.

Calculating the Common Ratio

• ⚡ By examining the transition from 51 to 153, it's observed that multiplying by 3 yields the next term (51 * 3 = 153).
• ✅ This pattern is confirmed for subsequent terms: 153 * 3 = 459, and 459 * 3 = 1377, establishing the common ratio as 3.

Determining the Initial Term (f(0) or a sub 0)

• 💡 To find the term before the first given term (51), we reverse the multiplication by dividing by the common ratio: 51 / 3 = 17.
• 📌 This value, 17, represents the initial term, which can be denoted as f(0) or a sub 0.

Recursive Formula Forms

• ✍️ The recursive formula can be expressed using function notation: f(n) = 3 * f(n-1), with the initial condition f(0) = 17.
• 📝 Alternatively, using subscript notation: a sub n = 3 * a sub(n-1), with the initial condition a sub 0 = 17.