Recursive Definition of Sequence: 6, 12, 24, 48

Math Problem Statement

Find the recursive definition of this sequence. 6, 12, 24, 48

The given sequence is $6, 12, 24, 48$ . Let's determine its recursive definition.

Look for the pattern:
To find the recursive relation, we compare successive terms:
- $12 \div 6 = 2$
- $24 \div 12 = 2$
- $48 \div 24 = 2$
  This shows that each term is double the previous term.
Write the recursive formula:
The formula can be expressed as:
$a_n = 2a_{n-1}, \quad \text{with } a_1 = 6$
Validate:
- $a_2 = 2a_1 = 2 \times 6 = 12$
- $a_3 = 2a_2 = 2 \times 12 = 24$
- $a_4 = 2a_3 = 2 \times 24 = 48$
  The formula works.

The recursive definition of the sequence is:

a_n &= 2a_{n-1}, \\ a_1 &= 6 \end{aligned}}$$ This matches the third option provided in the image. Do you want further clarification or details? --- ### Related Questions: 1. What is the difference between a recursive and an explicit formula for a sequence? 2. Can you generalize this sequence to find the explicit formula? 3. How would the recursive formula change if the sequence were $$6, 18, 54, \ldots$$? 4. What are common applications of recursive sequences in real-world problems? 5. How would the recursive formula look if the first term were changed to $$a_1 = 3$$? ### Tip: When working with recursive sequences, always validate by substituting terms to ensure the formula holds!

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Sequences
Recursive Formulas

a_n = 2a_{n-1}, a_1 = 6

Arithmetic and Geometric Sequences

Grades 6-8

Recursive Formula for Sequence 24, 12, 6, ...

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Recursive Formula for the Geometric Sequence 8, 48, 288, 1728

Explicit Definition of Geometric Sequence 8, 24, 72, 216