Arithmetic Sequences – Comprehensive Study Notes

Objectives

  • Define and describe three special kinds of sequences: arithmetic, geometric, and harmonic.
  • Solve problems that involve these sequences, with emphasis in this lesson on arithmetic sequences.

Key Definitions & Principles

  • Sequence: An ordered list of numbers written according to a rule.
  • Arithmetic Sequence (A.S.)
    • A sequence in which a constant value d (called the common difference) is added to each term to obtain the next term.
    • Symbolically, if {a1,a2,a3,\dots} is an arithmetic sequence, then a{n}=a_{n-1}+d for n\ge 2.
  • First Term: a1 (also written \alpha1 in the slides).
  • Common Difference: d=a2-a1=a3-a2=\dots (can be positive, negative, or zero).
  • General or nth‐Term Formula:
    • an=a1+(n-1)d.
    • Enables computation of any term without listing all preceding terms.

Priming Activity (Warm-Up Sequences)

  • (a) 2,\;5,\;8,\;11,\;14 → A.S. with d=3.

  • (b) 10,\;20,\;30,\;40,\;50 → A.S. with d=10.

  • (c) 7,\;4,\;1,\;-2,\;-5 → A.S. with d=-3 (decreasing sequence).

  • (d) 1,\;4,\;9,\;16,\;25 → Not arithmetic (perfect squares, actually quadratic growth).

  • (e) 100,\;95,\;90,\;85,\;80 → A.S. with d=-5.

    Interpretation & Pedagogical Notes:

    • Warm-up trains students to recognize constant addition vs other patterns.
    • Identifying d quickly is crucial for more advanced tasks (e.g.
      finding distant terms or solving for n).

Step-by-Step Examples & Detailed Solutions

Example 1 – Forward Computation (constructing the sequence)

Sequence: 2,5,8,11,\dots

  • Given: a_1 = 2,\; d = 3.
  • Compute successive terms:
    • a2 = a1+d = 2+3 = 5
    • a3 = a2+d = 5+3 = 8
    • a4 = a3+d = 8+3 = 11
  • nth-term (general term): a_n = 2 + (n-1)\,3.

Example 2 – 16th Term of 1,5,9,13,\dots

  • Identify parameters: a_1 = 1,\; d = 4\;(9-5),\; n = 16.
  • Apply nth-term formula:
    a_{16}=1+(16-1)\,4=1+15\cdot4=1+60=61.
  • Interpretation: 16th term is quite large because positive d "pushes" the sequence upward linearly.

Example 3 – 20th Term of 25,23,21,19,17,\dots

  • a_1 = 25,\; d = -2,\; n = 20.
  • Calculation:
    a_{20}=25+(20-1)(-2)=25+19\cdot(-2)=25-38=-13.
  • Reflective Point: Negative d causes a linear decline; by the 20th term value becomes negative.

Example 4 – Identify a Specific Term Value (Backward Problem)

"In 50,45,40,35,\dots which term equals 5?"

  • a1 = 50,\; d = -5,\; an = 5.
  • Solve for n:
    \begin{aligned}
    5 &= 50+(n-1)(-5)\
    5 &= 50-5(n-1)\
    5 &= 50-5n+5\
    5n &= 50\
    n &= 10.
    \end{aligned}
  • Answer: 5 is the 10th term (a_{10}).

Example 5 – Another Backward Problem

"In 7,10,13,16,\dots which term equals 43?"

  • a1 = 7,\; d = 3,\; an = 43.
    \begin{aligned}
    43 &= 7+(n-1)3\
    43 &= 7+3n-3\
    3n &= 39\
    n &= 13.
    \end{aligned}
  • Conclusion: 43 is the 13th term.

Example 6 – Finding a_1 and d from Two Known Terms

"4th term is 34 and 10th term is 22. Find a1, d, and an."

  1. Use the nth-term formula for both positions:

    • a4 = a1+3d = 34
    • a{10}=a1+9d = 22

  2. Subtract equations (eliminate a1): (a1+9d)-(a_1+3d)=22-34 \;\Rightarrow\; 6d=-12 \;\Rightarrow\; d=-2.

  3. Solve for a1 using a4:
    34=a1+3(-2) \;\Rightarrow\; 34=a1-6 \;\Rightarrow\; a_1=40.

  4. General term:
    a_n=40+(n-1)(-2)=40-2n+2=42-2n. (Slide showed -2n+42 which is algebraically identical.)

    Observations & Extensions:

    • Distance between indices (10th–4th = 6 steps) multiplied by d equals difference in values (22–34 = –12).
    • Shows linear nature: every 6 steps, value drops by 6d.

Compact Formula Summary

  • nth term: an=a1+(n-1)d.
  • Common difference: d=a{n}-a{n-1}.
  • To find n when an is known: n=\frac{an-a_1}{d}+1 (provided d\neq 0).

Problem-Solving Strategies & Tips

  • Always identify a_1 and d before anything else.
  • Watch sign of d (positive → increasing, negative → decreasing).
  • Use nth-term formula forward (find a_n) or backward (solve for n).
  • When two non-consecutive terms are known, set up a system or exploit difference: aj-ai=(j-i)d. Quick way to extract d.
  • Verify arithmetic vs non-arithmetic early—saves misapplication of formulas.

Connections & Broader Significance

  • Arithmetic sequences model linear growth or decline in finance (e.g.
    equal wage increases), physics (uniform motion with constant velocity), and computing (memory addresses with fixed strides).
  • Prepares groundwork for arithmetic series (sum of A.S.) and comparison to geometric sequences (exponential change) and harmonic sequences (reciprocals of arithmetic sequences).
  • Ethically, selecting correct mathematical model is vital—overusing linear projections may misrepresent exponential realities (e.g.
    disease spread, compound interest).

Quick Reference (Checklist)

  • Identify: constant difference? Yes → arithmetic.
  • Parameters: write a1, d, n, an.
  • Choose formula orientation (find term or locate index).
  • Substitute carefully; include parentheses around n-1 when d is negative.
  • Simplify & verify: plug result back if time permits.