Arithmetic Sequences – Comprehensive Study Notes

Learning Targets

• After studying this module you should be able to:
  • Define an arithmetic sequence.
  • Identify every element of an arithmetic sequence (first term, common difference, nth term, number of terms).
  • Compute the common difference, missing terms, specific nth terms, and locate a term’s position in the sequence.
  • Apply the general term formula $(a<em>n = a</em>1 + (n-1)d)$ in diverse problem settings, including practical scenarios (e.g., a weekly savings plan).

Definition of Arithmetic Sequence

• An arithmetic sequence is a list of numbers in a specific order where each successive term is obtained by adding a constant number called the common difference ($d$).
• Symbolically:
  • If $(a<em>1, a</em>2, a<em>3, \dots)$ is an arithmetic sequence, then $a</em>{k+1} = a_k + d$ for every integer $k \ge 1$.
• Essential Vocabulary:
  • Term ($a_n$): an individual entry in the sequence, where $n$ is its position.
  • First term ($a_1$): the initial value.
  • Common difference ($d$): a fixed value that is repeatedly added (can be positive, negative, or zero).
  • nth term / general term ($a_n$): the expression that gives any term as a function of $n$.

Common Difference

• Operational definition: the amount added to each term to generate the next term.
• Computation rule: subtract a term from the term that follows it.
  • $d = a<em>{k+1} - a</em>k$
• Properties:
  • $d$ can be positive (sequence rises), negative (sequence falls), or zero (constant sequence).
  • Knowing $d$ and any single term permits reconstruction of the entire sequence.

Finding the Common Difference (Illustrations)

• Example 1: 5,\;8,\;11,\;14,\;__,\;__,\;__
  • $d = 8-5 = 3 = 11-8 = 14-11$.
  • Extension: $17,\;20,\;23$ (continue adding $+3$).
• Example 2: 5,\;1,\;-3,\;-7,\;__,\;__,\;__
  • $d = 1-5 = -4 = (-3)-1 = (-7)-(-3)$.
  • Extension: $-11,\;-15,\;-19$ (continue adding $-4$).

General Term (nth-Term) Formula

• Fundamental relation: $\boxed{\displaystyle a<em>n = a</em>1 + (n-1)d}$.
• Variable meanings:
  • $a_n$ – value of the nth term required.
  • $a_1$ – first term (anchor of the sequence).
  • $n$ – ordinal position of the queried term.
  • $d$ – common difference.
• Manipulations:
  • To find a missing term (unknown $a<em>n$): substitute known $a</em>1,\ d,\ n$.
  • To locate the position of a known value (unknown $n$): solve the linear equation for $n$.
  • To uncover $d$ when two distinct terms are known: rearrange to isolate $d$.

Real-World Example (Savings Plan)

• Scenario: You save ₱50 (week 1), ₱65 (week 2), ₱80 (week 3)…
  • Identify $a_1 = 50$, $d = 65 - 50 = 15$.
  • Required: amount in week 10 $\,(n = 10)$.
  • Compute: $a_{10} = 50 + (10-1)\cdot15 = 50 + 135 = 185$.
  • Interpretation: You will save ₱185 in the 10th week.

Worked Examples (from slides)

Example 1 – 32nd Term

• Sequence: $79,\;75,\;71,\;67,\;63,\dots$
• Given: $a_1 = 79,\ d = 75-79 = -4,\ n = 32$.
• Solution:
  • $a_{32} = 79 + (32-1)(-4) = 79 + 31(-4) = 79 - 124 = -45$.
• Result: $a_{32} = -45$.

Example 2 – Position of a Specific Term

• Same sequence: $79,\;75,\;71,\;\dots$ Which term equals $-169$?
• Known: $a<em>1 = 79,\ d = -4,\ a</em>n = -169$.
• Compute $n$:
  • $-169 = 79 + (n-1)(-4)$
  • $-169 - 79 = -4(n-1) \Rightarrow -248 = -4(n-1)$
  • Divide: $62 = n-1 \Rightarrow n = 63$.
• Conclusion: $-169$ is the 63rd term.

Example 3 – Finding d

• Data: $a<em>1 = 118,\ a</em>{31} = -512,\ n = 31$.
• Using the formula:
  • $-512 = 118 + (31-1)d$
  • $-630 = 30d$
  • $d = -21$.
• Interpretation: The sequence decreases by $21$ each step.

Example 4 – Finding the First Term

• Given: $d = 3,\ a_{564} = 1697,\ n = 564$.
• Solve for $a_1$:
  • $1697 = a_1 + (564-1)\cdot3$
  • $1697 = a_1 + 1689$
  • $a_1 = 1697 - 1689 = 8$.
• First term: $8$.

Example 5 – Using Two Interior Terms

• Known terms: $a<em>{18} = 61,\ a</em>{12} = 19$; Find $a_{24}$.
• Step 1: Find $d$ using $d = \dfrac{a<em>y - a</em>x}{y-x}$.
  • $d = \dfrac{61 - 19}{18 - 12} = \dfrac{42}{6} = 7$.
• Step 2: Find $a<em>1$ with $a</em>{18}$:
  • $61 = a<em>1 + 17(7) = a</em>1 + 119 \Rightarrow a_1 = 61 - 119 = -58$.
• Step 3: Compute $a_{24}$:
  • $a_{24} = -58 + (24-1)\cdot7 = -58 + 23\cdot7 = -58 + 161 = 103$.
• Result: $a_{24} = 103$.

Concept Reinforcement – “Truth or Trick” Summary

• Claim 1: "In an arithmetic sequence, each term increases or decreases by the same amount." → Truth (definition of $d$).
• Claim 2: "The formula for the nth term is $a<em>n = a</em>1 + (n-1)d$." → Truth (general term).
• Claim 3: "The sequence 3, 6, 9, 12 is not an arithmetic sequence." → Trick (it actually is, with $d = 3$).
• Claim 4: "The common difference can be negative." → Truth (Example 2, $d=-4$).
• Claim 5: "In an arithmetic sequence, you need the first term and common difference to find any term." → Truth (general term requires $a_1, d$).

Key Takeaways & Connections

• Arithmetic sequences model linear growth or decay; the common difference is analogous to the slope in the line $y = mx + b$.
• Practical applications include savings plans, installment payments, predictable scheduling, depreciation, and number patterns.
• Algebraically rearranging $a<em>n = a</em>1 + (n-1)d$ lets us solve for any one of the four variables if the other three are known.
• Ethical implications: When dealing with financial sequences (e.g., loans), understanding linear accumulation prevents misinformation and promotes informed decision-making.