Curriculum
Login

Curriculum

Mathematics

Unit 1

0/12

Unit 2

0/14

Unit 3

0/13

Unit 4

0/8

Unit 5

0/12

Unit 6

0/8

Unit 7

0/10
Text lesson

Lesson 1 – Arithmetic and Geometric Sequences and Series

Mathematics

Lesson 1 – Arithmetic and Geometric Sequences and Series

Overview

Overview

🌳 Concept tree

  • Lesson 1 – Arithmetic and Geometric Sequences and Series

├── Key terms

├── Laws and key relationships

  • ├── Sequence Foundations and Theory of Arithmetic Progressions (A.P.)
  • ├── Theory of Geometric Progressions (G.P.) and Convergence to Infinity
  • ├── Sigma Notation (∑) and Analytical Summation Theorems

├── Worked examples

├── Exam tips

└── Real-world applications

Lesson 1 – Arithmetic and Geometric Sequences and Series · 1 / 9
Key terms

Key terms

Comparison aspectMain ideaKeywords
Sequence
  • A function whose domain is the set of positive integers ℤ+ = {1
  • …} and whose range is a subset of real numbers ℝ. Its terms are denoted T1
functionwhosedomainpositive
Arithmetic Sequence (A.P.)A sequence in which the difference between any term and its immediate predecessor is a constant real number d: T_(n+1) – Tn = d for all n ∈ ℤ+.differencebetweentermimmediate
Geometric Sequence (G.P.)A sequence in which the ratio of any term to its immediate predecessor is a constant non-zero real number r: T_(n+1) / Tn = r for all n ∈ ℤ+.ratiotermimmediatepredecessor
Arithmetic Mean
  • The arithmetic mean of two real numbers a and b is (a + b) / 2. If three numbers form an A.P. (a
  • then b = (a + c) / 2
realnumbersthreeform
Geometric Mean
  • The geometric mean of two numbers a and b of the same sign is ±√(ab). If three numbers form a G.P. (a
  • then b2 = ac
numberssamesignthree
Lesson 1 – Arithmetic and Geometric Sequences and Series · 2 / 9
Laws and key relationships

Laws and key relationships

Comparison aspectMain ideaKeywords
General Term of an Arithmetic Progression[ Formula: Tn = a + (n – 1)d ] Where a is the first term, d is the common difference, and n is the term rank (n ∈ ℤ+).Formulafirstcommondifference
Sum of First n Terms of an Arithmetic Progression[ Formula: Sn = (n / 2)[2a + (n – 1)d] = (n / 2)[a + l] ] Where l is the last term (l = Tn). Represents the area under the linear sequence staircase.FormulalastRepresentsarea
General Term and Infinite Sum of a Geometric Progression[ Formula: Tn = a · rn – 1; S_∞ = a / (1 – r) (valid if and only if |r| < 1) ] If the common ratio satisfies -1 < r < 1, the infinite geometric series converges to a finite limiting value S_∞.Formulavalidcommonratio
Lesson 1 – Arithmetic and Geometric Sequences and Series · 3 / 9
Sequence Foundations and Theory of Arithmetic Progressions (A.P.)

Sequence Foundations and Theory of Arithmetic Progressions (A.P.)

  • A sequence is formally defined as a mathematical mapping f ℤ+ → ℝ.
  • Unlike continuous real functions defined on intervals, a sequence consists exclusively of discrete, ordered elements.
  • The functional value f(n) is universally designated as the n-th term, denoted Tn.
Arithmetic and Geometric Sequences: a grandstand with rows of 10, 12, 14 seats and so on shows arithmetic and geometric sequences and series: the 10th row has 28 and the total is 190.
Arithmetic and Geometric Sequences: a grandstand with rows of 10, 12, 14 seats and so on shows arithmetic and geometric sequences and series: the 10th row has 28 and the total is 190.
  • An Arithmetic Progression (A.P.) is defined by the invariant linear recurrence relation: T_(n+1) – Tn = d (constant common difference).
  • If d > 0, the progression is strictly increasing; if d < 0, it is strictly decreasing; if d = 0, it is a constant sequence.
  • General Term Formula By mathematical induction, T1 = a, T2 = a + d, T3 = a + 2d, yielding the universal general term: Tn = a + (n – 1)d. Notice that Tn is a linear polynomial in n whose leading coefficient equals d.
  • Summation of an A.P. (Gauss Formula): The sum Sn = T1 + T2 + … + Tn of the first n terms can be written forwards and backwards, summing pairs to yield: 2 Sn = n(a + l) → Sn = (n / 2)[a + l] = (n / 2)[2a + (n – 1)d], where l = Tn is the last term.
  • Arithmetic Means When k terms are inserted between two numbers A and B to form an A.P., the total number of terms becomes n = k + 2. The common difference is: d = (B – A) / (k + 1). The first inserted mean is T2 = A + d, and the last inserted mean is T_(n-1) = B – d.
Lesson 1 – Arithmetic and Geometric Sequences and Series · 4 / 9
Theory of Geometric Progressions (G.P.) and Convergence to Infinity

Theory of Geometric Progressions (G.P.) and Convergence to Infinity

  • A Geometric Progression (G.P.) is characterized by an invariant multiplicative factor: T_(n+1) / Tn = r (constant common ratio, r ≠ 0).
  • General Term: T1 = a, T2 = ar, T3 = ar2, giving: Tn = a · rn – 1.
  • Partial Sum Formula Multiplying the partial sum Sn = a + ar + ar2 + … + arn−1 by r and subtracting gives: Sn – r Sn = a – a rn → Sn = [a(1 – rn)] / (1 – r) = (a – l r) / (1 – r), where r ≠ 1.
  • Infinite Series & Convergence What happens when the number of terms n approaches infinity? If |r| < 1 (i.e., -1 < r < 1), the term rn decays asymptotically to zero: limn -> ∞ rn = 0. Therefore, the infinite sum converges to: S_∞ = limn -> ∞ Sn = a / (1 – r). If |r| ≥ 1, rn diverges, and the sum to infinity does NOT exist.
Geometric Means→Inserting k geometric means between A and B produces a G.P. of n = k + 2 terms. Common ratio: r = (B / A)1 / (k + 1)).
Lesson 1 – Arithmetic and Geometric Sequences and Series · 5 / 9
Sigma Notation (∑) and Analytical Summation Theorems

Sigma Notation (∑) and Analytical Summation Theorems

Comparison aspectMain ideaKeywords
•The Greek capital letter Sigma (∑) is used to denote series sums compactly: ∑_{r=1}n Tr = T1 + T2 + … + Tn.GreekcapitalletterSigma
Analytical Summation Theorems
  • (1) Sum of a constant: ∑{r=1}n c = c × n. (2) Linearity: ∑{r=1}n [c · f(r)] = c · ∑_{r=1}n f(r)
  • and ∑ [f(r) ± g(r)] = ∑ f(r) ± ∑ g(r)
constantLinearity
(3) Sum of the first n positive integers∑{r=1}n r = 1 + 2 + … + n = [n(n + 1)] / 2. (4) Sum of the first n squares: ∑{r=1}n r2 = 12 + 22 + … + n2 = [n(n + 1)(2n + 1)] / 6squares
Index Shift Property
  • To evaluate ∑{r=k}n Tr where k > 1
  • apply: ∑{r=k}n Tr = ∑{r=1}n Tr – ∑{r=1}k-1 Tr
evaluateapply
Lesson 1 – Arithmetic and Geometric Sequences and Series · 6 / 9
Worked examples

Worked examples

✍️ Worked examples

Comparison aspectMain ideaKeywords
Worked Example 1Finding the Maximum Partial Sum of an Arithmetic SequenceFindingMaximumPartialArithmetic
Problem Statement
  • Given the arithmetic progression (35
  • …): (1) Find the number of positive terms
  • (2) Calculate the maximum sum of this sequence and state how many terms must be taken to achieve it
GivenarithmeticprogressionFind
Detailed Solution & Derivation
  • Step 1: First term a = 35. Common difference d = 31 – 35 = -4. Step 2: To find the number of positive terms
  • solve Tn > 0: a + (n – 1)d > 0 → 35 + (n – 1)(-4) > 0 → 35 – 4n + 4 > 0 → 39 > 4n → n < 39/4 = 9.75. Because n ∈ ℤ+
  • the sequence has exactly 9 positive terms (T9 is positive
  • T10 is negative). Step 3: The maximum sum occurs when adding all positive terms (adding negative terms would reduce the sum). Therefore
  • n = 9 terms give the maximum sum. Step 4: Calculate S9: S9 = (9/2)[2(35) + (9 – 1)(-4)] = 4.5 [70 + 8(-4)] = 4.5 [70 – 32] = 4.5 × 38 = 171. (Note: S10 gives 171 + (-1) = 170)
StepFirsttermCommon
✍️ Worked examples→Geometric Sequence Formulation from Conditions
Comparison aspectMain ideaKeywords
Problem Statement
  • A geometric sequence with positive terms has its second term equal to 6
  • and the sum of its third and fourth terms equals 36
geometricsequencepositiveterms
•Find the sequence and calculate the sum of its first 6 termsFindsequencecalculatefirst
Detailed Solution & Derivation
  • Step 1: T2 = a·r = 6 → a = 6/r. Step 2: T3 + T4 = a·r2 + a·r3 = 36 → a·r2(1 + r) = 36. Step 3: Substitute a = 6/r: (6/r)·r2(1 + r) = 36 → 6r(1 + r) = 36 → r2 + r – 6 = 0. Step 4: Factor: (r + 3)(r – 2) = 0. Since terms are positive
  • r > 0
  • so r = 2 (reject r = -3). Step 5: a = 6/2 = 3. The sequence is: (3
  • …). Step 6: Sum of first 6 terms: S6 = [a(r6 – 1)] / (r – 1) = [3(26 – 1)] / (2 – 1) = 3(64 – 1) = 3 × 63 = 189
StepSubstituteFactorSince
Lesson 1 – Arithmetic and Geometric Sequences and Series · 7 / 9
Exam tips

Exam tips

Comparison aspectMain ideaKeywords
Exam SecretTo find the rank of the first negative term in an A.P., solve Tn < 0. The first integer strictly greater than the resulting decimal fraction is the rank.findrankfirstnegative
Crucial PropertyIn any A.P., Tn = Sn – S_(n-1) for all n ≥ 2. For n = 1, T1 = S1.
•
  • Remember that an alternating geometric sequence (r < 0) oscillates between positive and negative values. If |r| < 1
  • it still converges to S_∞ = a / (1 – r)
Rememberalternatinggeometricsequence
Lesson 1 – Arithmetic and Geometric Sequences and Series · 8 / 9
Real-world applications

Real-world applications

Comparison aspectMain ideaKeywords
Financial Mathematics & Compound InterestBank investments compounding annually at interest rate i form a geometric progression: An = P(1 + i)n.Bankinvestmentscompoundingannually
Nuclear Chain Reactions
  • In uranium fission
  • each neutron generates 2 or 3 secondary neutrons
  • causing nuclear power output to expand geometrically
uraniumfissionneutrongenerates
Depreciation of Capital AssetsEquipment book values diminishing at a constant annual depreciation percentage d follow a geometric decay curve: Vt = V0(1 – d)t.Equipmentbookvaluesdiminishing
•
  • egp-ex{margin:18px 0
  • border:1px solid rgba(79
  • 224
  • border-radius:16px
  • overflow:hidden
  • background:#fff}.egp-ex-h{padding:10px 16px
  • font-weight:800
  • background:linear-gradient(90deg
  • #4f3fe0
  • #22c1c3)
  • color:#fff}.egp-ex-tabs{display:flex
  • flex-wrap:wrap
  • gap:6px
  • padding:10px 12px
  • background:#f7f6ff}.egp-ex-b{border:1px solid rgba(79
  • 224
  • background:#fff
  • border-radius:20px
  • padding:5px 14px
  • font-weight:700
  • cursor:pointer}.egp-ex-b.on{background:#4f3fe0
  • color:#fff}.egp-ex-p{padding:12px 16px}
margin18pxbordersolid
Lesson 1 – Arithmetic and Geometric Sequences and Series · 9 / 9