Arithmetic & Geometric Sequences – Comprehensive Study Notes

Definition & Classification of Sequences

  • Sequence – an ordered list of numbers written according to a specific rule.

  • Two major families stressed in the transcript:

    • Arithmetic Sequence (A.S.) – a constant value is added to each term to obtain the next.

    • Constant increment is called the common difference d.

    • General term (nth term): an=a1+(n-1)d

    • Sum of the first n terms: Sn=\dfrac{n}{2}\bigl(a1+an\bigr) or Sn=\dfrac{n}{2}\bigl(2a_1+(n-1)d\bigr)

    • Geometric Sequence (G.S.) – each term is obtained by multiplying the previous one by a fixed number.

    • Constant multiplier is the common ratio r.

    • General term: an=a1\,r^{\,(n-1)}

    • Sum of first n terms when |r|\neq 1: Sn=\dfrac{a1\bigl(r^{\,n}-1\bigr)}{r-1}

Core Arithmetic-Sequence Content

  • Distinguishing property: d=a{k+1}-ak\;\;(\text{constant for all }k).

  • Examples mentioned (all have identical first–difference tables of constant d):

    • 2,3,4,5,\ldots ,10 (common difference d=1)

    • 7,10,13,16,19,22,25 (common difference d=3)

    • 4,9,14,19,\ldots (common difference d=5)

    • 2,5,8,11\,(\ldots) (common difference d=3)

    • 17,12,7,2,\ldots (common difference d=-5 – illustrates negative d)

  • Finding a specific term – worked computations shown:

    • For 1,5,9,13,\ldots
      d=4, \; a{16}=a1+(16-1)d = 1+15\times4 = 61

    • Sequence 11,5,-1,\ldots ( d=-6 ).
      a_{60}=11+(60-1)(-6)=11-354=-343

  • Inserting arithmetic means:

    • Between 9 and 33 with two A.M.s — full set 9,17,25,33 where d=8.

    • Between -12 and 13 with four A.M.s — final complete list -12,-7,-2,3,8,13 (still d=5)

Sum of an Arithmetic Sequence – Transcript Examples

  • 3,6,9,12

    • a1=3,\, a4=12,\,n=4
      S_4=\dfrac{4}{2}(3+12)=30

  • 20-term set beginning 15,19,23,\ldots ,91 (clearly d=4)

    • a1=15,\, a{20}=91
      S_{20}=\dfrac{20}{2}(15+91)=10\times106=1060

Core Geometric-Sequence Content

  • Defining property: r=\dfrac{a{k+1}}{ak}\;\;(\text{constant}).

  • Transcript data set used for practice: -2,-6,-18,-54,\ldots

    • r=3 (still negative because of the first negative term).

    • Individual terms re-derived:
      a5=-162,\;a6=-486,\;a7=-1458,\;a8=-4374 (via a_n=-2\,3^{n-1}).

    • General reminder formula repeated: an=a1\,r^{n-1}.

  • Other pure-ratio examples cited: 2,6,18,54\,(r=3) and 3,9,27,81,\ldots ,729\,(r=3).

Quadratic (Second-Difference Constant) Sequences

  • If 1st differences vary but 2nd differences are constant, the general term is quadratic.

  • Transcript illustration: 2,5,10,17,26\,\ldots

    • 1st diff: +3,+5,+7,+9
      2nd diff: always +2 → confirming quadratic.

    • Proposed formula: a_n=n^2+1 (check: n=4 \Rightarrow 17 etc.)

Linear-Formula Recognition Practice

  • From the set 2,7,12,17,\ldots first term a_1=2 and constant diff d=5.

    • Equivalent closed form advertised in notes: a_n=5n-3.

Fractional Sequences – Building Rational General Terms

  • Demonstrated approach: "separate numerator & denominator".

  • One activity reaches the general form
    a_n=\dfrac{2n-1}{2n} → explicit terms \tfrac12,\tfrac34,\tfrac56,\tfrac78,\ldots

  • Technique recap:

    • Identify denominator pattern ( here 2n ) & numerator pattern ( 2n-1 ).

    • Check first three written terms against those formulas.

Sequences with Radicals (Roots)

  • Idea: treat the coefficient in front of the radical as the pattern.

  • Transcript sketches:

    • \sqrt3,\,2\sqrt3,\,3\sqrt3,\,\ldots has rule a_n=n\sqrt3.

    • A slightly shifted variant a_n=(2n-1)\sqrt3 would begin \sqrt3,3\sqrt3,5\sqrt3,\ldots.

  • Same reasoning used to suggest pattern a_n=n\sqrt5 from noted raw data \sqrt5,2\sqrt5,3\sqrt5,\ldots.

General Procedural Skills Emphasised

  • To decide arithmetic vs geometric:

    1. Compute two successive first differences; if equal ⇒ arithmetic.

    2. Compute two successive ratios; if equal ⇒ geometric.

    3. If neither behaves, probe 2nd differences ⇒ quadratic, or analyze numerator/denominator or radical coefficient.

  • To find an unknown term of an A.S. or G.S.:

    1. Derive d or r.

    2. Plug into the general term formula with desired n.

  • For sum tasks in A.S.:

    • Decide whether you know an; if not, compute it first, then substitute into Sn=\dfrac{n}{2}(a1+an).

Mini-Glossary/Reminders

  • Common difference (d) – fixed additive change in an A.S.

  • Common ratio (r) – fixed multiplicative change in a G.S.

  • Arithmetic Mean (A.M.) between terms p and q: \tfrac{p+q}{2} (when a single mean is inserted).

  • k Arithmetic means between p and q leads to k+2 total terms, d=\dfrac{q-p}{k+1}.

  • Second difference – difference of successive first differences; constant ⇒ quadratic rule a_n=An^2+Bn+C.

Complete List of Sequences Explicitly Written in Transcript (Classified)

  • Arithmetic:
    2,3,4,5,\ldots,10 | 7,10,13,16,19,22,25 | 4,9,14,19,\ldots | 2,5,8,11,\ldots | 17,12,7,2,\ldots | 1,5,9,13,\ldots | 9,17,25,33 | -12,-7,-2,3,8,13 | 2,7,12,17,\ldots

  • Geometric:
    2,6,18,54 | 3,9,27,81,\ldots,729 | -2,-6,-18,-54,\ldots

  • Quadratic (2nd diff =2): 2,5,10,17,26,\ldots

  • Fraction sequence prototype: \dfrac{1}{2},\dfrac{3}{4},\dfrac{5}{6},\dfrac{7}{8},\ldots

  • Radical prototypes: \sqrt3,2\sqrt3,3\sqrt3,\ldots and \sqrt5,2\sqrt5,3\sqrt5,\ldots

These bullet-pointed notes encompass every explicit numeric example, all core formulas (rendered in LaTeX), and the conceptual / procedural commentary scattered across the seven pages of the transcript. Use them as a standalone, exam-ready reference on arithmetic, geometric, and related special sequences.