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 ${a<em>1,a</em>2,a<em>3,\dots}$ is an arithmetic sequence, then $a</em>{n}=a_{n-1}+d$ for $n\ge 2$.
• First Term: $a<em>1$ (also written $\alpha</em>1$ in the slides).
• Common Difference: $d=a<em>2-a</em>1=a<em>3-a</em>2=\dots$ (can be positive, negative, or zero).
• General or nth‐Term Formula:
  • $a<em>n=a</em>1+(n-1)d.$
  • 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:
  • $a<em>2 = a</em>1+d = 2+3 = 5$
  • $a<em>3 = a</em>2+d = 5+3 = 8$
  • $a<em>4 = a</em>3+d = 8+3 = 11$
• 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?"

• $a<em>1 = 50,\; d = -5,\; a</em>n = 5.$
• 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?"

• $a<em>1 = 7,\; d = 3,\; a</em>n = 43.$
  \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 $a<em>1$, $d$, and $a</em>n$."

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

   • $a<em>4 = a</em>1+3d = 34$
   • $a<em>{10}=a</em>1+9d = 22$

2. Subtract equations (eliminate $a<em>1$): $(a</em>1+9d)-(a_1+3d)=22-34 \;\Rightarrow\; 6d=-12 \;\Rightarrow\; d=-2.$

3. Solve for $a<em>1$ using $a</em>4$:
   $34=a<em>1+3(-2) \;\Rightarrow\; 34=a</em>1-6 \;\Rightarrow\; a_1=40.$

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: $a<em>n=a</em>1+(n-1)d.$
• Common difference: $d=a<em>{n}-a</em>{n-1}.$
• To find $n$ when $a<em>n$ is known: $n=\frac{a</em>n-a_1}{d}+1$ (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: $a<em>j-a</em>i=(j-i)d.$ 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 $a<em>1$, $d$, $n$, $a</em>n$.
• 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.