How to Write Recursive Formulas for Arithmetic Sequences | Khan Academy

Identifying the Pattern in an Arithmetic Sequence

• 🎯 The video demonstrates how to find the recursive formula for a sequence when given terms 3 through 6.
• 🔍 To determine if the sequence is arithmetic or geometric, we examine the difference between consecutive terms.
• 📊 By converting all given terms to a common denominator (fourths), it becomes clear that the sequence is increasing by 3/4s at each step.

Constructing the Recursive Formula

• ✍️ The recursive formula defines each term based on the previous term.
• 💡 For an arithmetic sequence, the formula is expressed as f(n) = f(n-1) + d or a_n = a_{n-1} + d, where 'd' is the common difference.
• ➕ In this case, the common difference is 3/4s, leading to the formulas f(n) = f(n-1) + 3/4 and a_n = a_{n-1} + 3/4.

Determining the Initial Term

• 🔑 The problem provides terms 3 through 6, so the initial term (first term) needs to be calculated.
• ⏪ To find previous terms, we subtract the common difference (3/4s) from the known terms.
• 📉 Subtracting 3/4s from the third term (-12/4s) gives the second term (-15/4s).
• ⏪ Subtracting 3/4s again from the second term (-15/4s) yields the first term (-18/4s).
• ✅ The first term, -18/4s, simplifies to -9/2s, which can be represented as f(1) = -9/2 or a_1 = -9/2.