Lecture on Arithmetic Sequences

Overview

• Focus: Finding a formula for the nth term of an arithmetic sequence.
• Formula: $a_n = a_1 + (n - 1) \cdot d$
  • $a_1$ = first term
  • $n$ = term number
  • $d$ = common difference

Justification of the Formula

• Example Sequence: 3, 8, 13, 18, 23, 28
  • This is an arithmetic sequence with a common difference ($d$) of 5.
• Terms:
  • $a_1 = 3$
  • $a_2 = 3 + 5$
  • $a_3 = 3 + 2 \times 5$
  • $a_4 = 3 + 3 \times 5$
  • Pattern Recognition: Each term after the first can be calculated by adding the common difference multiplied by one less than the term number.
• General Formula: For $a_n$, use $a_1 + (n - 1) \cdot d$

Example Problems

Problem 1: Find the 10th Term

• Sequence: 3, 8, 13, 18, 23, 28, 33...
• Steps:
  1. Recognize $a_1 = 3$ and $d = 5$.
  2. Calculate: $a_{10} = 3 + (10 - 1) \cdot 5$
  3. Result: $a_{10} = 48$

Problem 2: Find the 202nd Term

• Steps:
  1. Use $a_{202} = 3 + (202 - 1) \cdot 5$
  2. Calculate: $3 + 201 \cdot 5 = 1008$

Problem 3: Find First Term and Common Difference

• Given: $a_8 = 25$, $a_{14} = 43$
• System of Equations:
  • $a_1 + 7d = 25$
  • $a_1 + 13d = 43$
• Solving System:
  1. Subtract equations to eliminate $a_1$.
  2. Solve for $d$: $6d = 18 \Rightarrow d = 3$
  3. Substitute back to find $a_1$: $a_1 = 4$

Conclusion

• Key Takeaways: Understanding the formula for the nth term and applying it to solve various problems.
• Challenges: Systems of equations can determine unknowns in arithmetic sequences.
• Further Questions: Encouragement to ask questions for clarification.