Sequences & Series – Comprehensive Study Notes

Basics of Sequences

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

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

• 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: $a<em>n = f\bigl(a</em>{n-1},a_{n-2},\ldots\bigr)$ 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. $a<em>{n-1}$ or $a</em>{n-2}$.

• Example 1
  $\begin{cases}a<em>1=-2\ a</em>n = n + 3\,a<em>{n-1}\end{cases}$ • Compute first five terms: – $a</em>1=-2$ (given)
  – $a<em>2 = 2 + 3(-2)= -4$ – $a</em>3 = 3 + 3(-4)= -9$
  – $a<em>4 = 4 + 3(-9)= -23$ – $a</em>5 = 5 + 3(-23)= -64$
  • Far-out terms (e.g. $a_{50}$) require stepping through all predecessors.

• Example 2 (two-term recursion)
  $\begin{cases}a<em>1=-1,\ a</em>2=1\ a<em>n = n + (n)\,a</em>{n-1} - (a_{n-2})\end{cases}$ (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 $a<em>{50}, a</em>{100},\ldots$

• Examples
  • $a<em>n = n^2 - 1$ → first three terms $0,3,8$. • Oscillating sign: $a</em>n =(-1)^{n-1}\,\dfrac{2n-1}{3n}$ → 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 $d$: $a<em>n = a</em>{n-1}+d$.

• Explicit form: $a<em>n = a</em>1 + (n-1)d$.

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

• Determine d quickly with two known terms:
  $d = \dfrac{a<em>j-a</em>i}{j-i}$.

Geometric Sequences

• Definition: Constant ratio $r$: $a<em>n = a</em>{n-1}\,r$.

• Explicit form: $a<em>n = a</em>1\,r^{\,(n-1)}$.

• 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
  $\begin{cases}a<em>1 = \dfrac25\ a</em>n = (-2)\,a_{n-1}\end{cases}$

• 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: $\sum<em>{k=1}^{n} a\,r^{k-1}$ vs $\sum</em>{k=0}^{n-1} a\,r^{k}$ 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:
  $S<em>n = \dfrac{n\bigl(a</em>1 + a<em>n\bigr)}{2}$ where • $n$ = number of terms, • $a</em>1$ = 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 $1\to10$: $S<em>{10}=\dfrac{10(1+10)}{2}=55$. • Any arithmetic series follows same template once $a</em>1,a_n,n$ 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.