Lecture Notes: Understanding Recursive Function and Arithmetic Sequences

Introduction to the Function

• The function F(n) is defined recursively as:
  • ( F(n) = F(n-1) + 6 )
  • This means each term is the previous term plus 6.
• Initial condition given:
  • ( F(1) = 8 )
• Important to establish an initial point for recursive definitions.

Calculating Terms of the Function

• For ( n = 1 ):
  • ( F(1) = 8 ) (given)
• For ( n = 2 ):
  • ( F(2) = F(1) + 6 = 8 + 6 = 14 )
• For ( n = 3 ):
  • ( F(3) = F(2) + 6 = 14 + 6 = 20 )
• For ( n = 4 ):
  • ( F(4) = F(3) + 6 = 20 + 6 = 26 )

Observing the Pattern

• The recursive function reveals an arithmetic sequence:
  • Start with 8, add 6 to each subsequent term.
• Pattern: Each term increases by 6.

Graphing the Sequence

• Graph the sequence on an N (horizontal) vs. F(N) (vertical) axis.
• Points to plot:
  • ((1, 8), (2, 14), (3, 20), (4, 26))
• The plotted points suggest a line.
• Conclusion:
  • While not continuous, the points align linearly.
  • Slope interpretation: Moving forward by 1 in N raises F(N) by 6.

Understanding Arithmetic Sequences

• Defined by a constant difference between consecutive terms.
• Linear Representation:
  • Points aligned on a line indicate an arithmetic sequence.
• Contrasts with Non-Linear Sequences:
  • A curved pattern indicates a non-arithmetic sequence.
• Key takeaway: Arithmetic sequences appear linear when graphed.

Final Thoughts

• Recognizing arithmetic sequences helps identify linear growth patterns.
• Recursive definitions require an initial condition for clarity and computation.