Sequences & Series – Comprehensive Study Notes

Basics of Sequences

• Sequence = ordered list of numbers separated by commas.
  • Each individual entry is a term.

• Symbol/Rule: an (read “a–sub–n”) denotes the n-th term or a formula that produces term n.
  Example: an = n^2 + 1.

• Representations
  • Raw list: 2,5,10,17,\ldots
  • Set/brace notation: {a_n} where braces indicate “this is a sequence, not a single number.”
  • Explicit formula inside braces signals a sequence (e.g. {n^2+1}).

• Term Indexing Trick: Write the indices 1,2,3,… above/below the values to spot relations (e.g. see that 1→2, 2→5, 3→10 all match n^2+1).

Strategies for Detecting Patterns

• Label indices to expose potential formulas.

• “Feel it” approach: guess → check → revise. There is no guaranteed hack beyond observation.

• Ask: Is there a constant difference (arithmetic)? constant ratio (geometric)? alternating sign? factorial? powers? etc.

Recursive Sequences

• Definition: Each term depends on one or more previous terms.
  General form: an = f\bigl(a{n-1},a_{n-2},\ldots\bigr) plus starting values.

• Mandatory components for a proper recursive definition:

  1. At least one initial term (often a_1).

  2. A rule containing an earlier term, e.g. a{n-1} or a{n-2}.

• Example 1
  \begin{cases}a1=-2\ an = n + 3\,a{n-1}\end{cases} • Compute first five terms: – a1=-2 (given)
  – a2 = 2 + 3(-2)= -4 – a3 = 3 + 3(-4)= -9
  – a4 = 4 + 3(-9)= -23 – a5 = 5 + 3(-23)= -64
  • Far-out terms (e.g. a_{50}) require stepping through all predecessors.

• Example 2 (two-term recursion)
  \begin{cases}a1=-1,\ a2=1\ an = n + (n)\,a{n-1} - (a_{n-2})\end{cases} (paraphrased from board).
  Calculated sequence: -1,\ 1,\ 2,\ 9,\ \ldots

Explicit (Nth-Term) Sequences

• Definition: Term expressed directly from index n; no earlier terms in the formula.

• Advantages: Jump straight to a{50}, a{100},\ldots

• Examples
  • an = n^2 - 1 → first three terms 0,3,8. • Oscillating sign: an =(-1)^{n-1}\,\dfrac{2n-1}{3n} → pattern ±, ±, ± … (the $(-1)^{n-1}$ piece forces the sign flips).
  • Alternate choices for sign-flippers: $(-1)^n,\ (-1)^{n+1},\ (-1)^{n+2}$ all yield ± sequence but start with opposite sign depending on exponent.

Comparing Recursive vs Explicit

Arithmetic Sequences

• Definition: Constant difference d: an = a{n-1}+d.

• Explicit form: an = a1 + (n-1)d.

• Identify by checking successive differences.
  • Example: -3,-1,1,3,\ldots → d=2.

• Determine d quickly with two known terms:
  d = \dfrac{aj-ai}{j-i}.

Geometric Sequences

• Definition: Constant ratio r: an = a{n-1}\,r.

• Explicit form: an = a1\,r^{\,(n-1)}.

• Tests
  • Multiply check: Does same factor carry you term-to-term?
  • Fraction/negative ratios allowed.

• Examples
  • 2,\ -4,\ 8,\ -16,\ldots → r=-2.
  • {-5,25,-125,625,\ldots} → r=-5 (simplify powers when exponents hide pattern).
  • \dfrac19\,3^{n} and \bigl(\dfrac23\bigr)^{n} are equivalent ways to write base·ratio forms.

Recursive vs Explicit in Practice

• Geometric recursive sample
  \begin{cases}a1 = \dfrac25\ an = (-2)\,a_{n-1}\end{cases}

• Geometric explicit version
  a_n = \dfrac25\,(-2)^{\,n-1}.

Sigma (Summation) Notation

• General: \displaystyle \sum_{k = \text{start}}^{\text{end}} f(k).
  • Bottom value = first index.
  • Top value = last index.
  • f(k) supplies the explicit term rule.

• Reading tips

  1. Plug successive k values: start, start+1, …, end.

  2. Write first three terms + last term to reveal pattern.

• Example 1 (not arithmetic/geometric)
  \sum_{k=1}^{n} (k+1)^2 → terms: (2)^2 + (3)^2 + (4)^2 + \cdots + (n+1)^2.

• Example 2 (alternating geometric)
  \sum_{k=0}^{\,n-1} (-1)^{k+1}\,2^{k} → after cleaning: -1 + 2 - 4 + \cdots + (-1)^{n}\,2^{\,n-1}.

• Shifting Indices
  • Starting at 1 vs 0 changes exponent offsets.
  • Example: \sum{k=1}^{n} a\,r^{k-1} vs \sum{k=0}^{n-1} a\,r^{k} are equivalent.

Writing a Series into Sigma Form

1. Decide whether first listed term is k=0 or k=1 (or other).

2. Identify explicit rule from last unsimplified term if shown.

3. Verify by back-substitution for first few k’s.

4. Adjust upper limit so final k reproduces final term.

Arithmetic Series (Sum of an Arithmetic Sequence)

• Historical story: Young Carl Gauss (late 1700s) summed 1+2+\dots+100 instantly (answer 5050) by pairing terms.

• Gauss pairing insight: first + last, second + second-last, … each pair gives same subtotal.

• Formula:
  Sn = \dfrac{n\bigl(a1 + an\bigr)}{2} where • n = number of terms, • a1 = first term,
  • a_n = last (n-th) term.

• Derivation mini-model using 1–10
  Pairs give 1+10 = 2+9 = 3+8 = 11. There are \frac{10}{2}=5 pairs ⇒ 5·11=55.

• Usage examples
  • Sum 1\to10: S{10}=\dfrac{10(1+10)}{2}=55. • Any arithmetic series follows same template once a1,a_n,n are known.

Key Insights & Tips

• Label indices; pattern often sits in “index → value” relationship.

• Recursive = dependent; explicit = independent of prior terms.

• Sign oscillations: multiply by $(-1)^n$‐style factor; shift exponent to set starting sign.

• Arithmetic → constant difference; geometric → constant ratio.

• Sigma notation: bottom = start index; top = stop index. Write ≥3 terms to check.

• In sigma, you may start at 0 or 1—adjust exponent or k-1 accordingly.

• When converting a raw series to sigma, copy the unsimplified last term; it usually reveals the rule.

• Gauss formula only works for arithmetic series; use geometric-series formulas (not covered here) for geometric sums.

Common Pitfalls

• Forgetting initial term in recursive definition → incomplete sequence.

• Mixing up n (index) with a_n (value).

• Using arithmetic formula on a geometric or non-linear sequence.

• Misplacing k start/stop numbers in sigma notation.

• Wrong exponent shift when starting sigma at 0 vs 1.

Vocabulary & Symbols

• a_n – nth term

• d – common difference (arithmetic)

• r – common ratio (geometric)

• S_n – sum of first n terms (series)

• \sum – summation symbol (sigma)

• Recursive keyword cues: “previous term,” “depends on,” a_{n-1}.

• Explicit cues: “nth term,” formula in n only.