Math Problem Statement

solve using formula above

solve using formula above

Solution

The given image illustrates the computation of F(4)F(4) using a recursive formula:

Recursive Formula

F(n)=2+F(n1),F(n) = 2 + F(n-1), with the base case: F(1)=2.F(1) = 2.

Explanation of F(4)F(4)

  • F(4)=2+F(3)F(4) = 2 + F(3)
  • F(3)=2+F(2)F(3) = 2 + F(2)
  • F(2)=2+F(1)F(2) = 2 + F(1)

Substituting the base case F(1)=2F(1) = 2:

  • F(2)=2+2=4F(2) = 2 + 2 = 4
  • F(3)=2+4=6F(3) = 2 + 4 = 6
  • F(4)=2+6=8F(4) = 2 + 6 = 8

Thus: F(4)=8.F(4) = 8.


Solve for F(3)F(3)

From the above process: F(3)=2+F(2),F(3) = 2 + F(2), where F(2)=4F(2) = 4. Therefore: F(3)=2+4=6.F(3) = 2 + 4 = 6.


If you have more questions or need detailed explanations, let me know!


Relative Questions:

  1. How does recursion simplify solving sequences like this?
  2. Can we generalize a formula for F(n)F(n) based on the recursive pattern?
  3. How is F(3)F(3) computed using recursion, step by step?
  4. What is the role of the base case F(1)=2F(1) = 2 in this computation?
  5. How does this recursive relationship relate to arithmetic progressions?

Tip:

When solving recursive problems, identify the base case first, as it anchors the recursion and prevents infinite loops.

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Math Problem Analysis

Mathematical Concepts

Recursion
Sequence Analysis
Arithmetic Progression

Formulas

F(n) = 2 + F(n-1)
Base Case: F(1) = 2

Theorems

Recursive Formula Analysis

Suitable Grade Level

Grades 8-10