Math Problem Statement

given the explicit formula below, write the recursive formula for the sequence g(n)=-12+6(n-1)

Solution

To convert the explicit formula g(n)=12+6(n1)g(n) = -12 + 6(n-1) into a recursive formula, we follow these steps:

Explicit Formula:

g(n)=12+6(n1)g(n) = -12 + 6(n-1)

Simplify: g(n)=12+6n6=6n18g(n) = -12 + 6n - 6 = 6n - 18

Recursive Formula:

A recursive formula defines each term based on the previous term, g(n1)g(n-1).

  1. From the explicit formula, observe that the difference between consecutive terms is constant: g(n)g(n1)=(6n18)[6(n1)18]=6g(n) - g(n-1) = (6n - 18) - [6(n-1) - 18] = 6 This is the common difference.

  2. Define the recursive relationship: g(n)=g(n1)+6,for n2g(n) = g(n-1) + 6, \quad \text{for } n \geq 2

  3. Include the base case (the first term of the sequence): g(1)=12g(1) = -12

Final Recursive Formula:

\begin{cases} -12, & \text{if } n = 1 \\ g(n-1) + 6, & \text{if } n \geq 2 \end{cases}$$ Would you like me to verify it further or illustrate the sequence? --- ### Related Questions: 1. What is the general difference between explicit and recursive formulas? 2. Can this sequence be represented as a geometric progression? 3. How can you graphically represent this arithmetic sequence? 4. What are the first 10 terms of the sequence, explicitly and recursively? 5. Can you derive the explicit formula from the recursive formula? ### Tip: For arithmetic sequences, always check the **common difference** when transitioning between recursive and explicit formulas.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Recursive Formulas
Explicit Formulas

Formulas

Explicit formula: g(n) = -12 + 6(n-1)
Recursive formula: g(n) = g(n-1) + 6, with g(1) = -12

Theorems

Arithmetic Sequence Formula

Suitable Grade Level

Grades 9-12