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🌳 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

Comparison aspect	Main idea	Keywords
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) – T_n = 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) / T_n = 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

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Laws and key relationships

Comparison aspect	Main idea	Keywords
General Term of an Arithmetic Progression	[ Formula: T_n = 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: S_n = (n / 2)[2a + (n – 1)d] = (n / 2)[a + l] ] Where l is the last term (l = T_n). Represents the area under the linear sequence staircase.	FormulalastRepresentsarea
General Term and Infinite Sum of a Geometric Progression	[ Formula: T_n = a · r^{n – 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.)

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 T_n.

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 S_n = T1 + T2 + … + Tn of the first n terms can be written forwards and backwards, summing pairs to yield: 2 S_n = n(a + l) → S_n = (n / 2)[a + l] = (n / 2)[2a + (n – 1)d], where l = T_n 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

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 · r^{n – 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)).

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Sigma Notation (∑) and Analytical Summation Theorems

Comparison aspect	Main idea	Keywords
•	The Greek capital letter Sigma (∑) is used to denote series sums compactly: ∑_{r=1}ⁿ T_r = 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)] / 6	squares
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

Comparison aspect	Main idea	Keywords
Worked Example 1	Finding the Maximum Partial Sum of an Arithmetic Sequence	FindingMaximumPartialArithmetic
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 aspect	Main idea	Keywords
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 terms	Findsequencecalculatefirst
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

Comparison aspect	Main idea	Keywords
Exam Secret	To find the rank of the first negative term in an A.P., solve T_n < 0. The first integer strictly greater than the resulting decimal fraction is the rank.	findrankfirstnegative
Crucial Property	In any A.P., T_n = S_n – 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

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Real-world applications

Comparison aspect	Main idea	Keywords
Financial Mathematics & Compound Interest	Bank investments compounding annually at interest rate i form a geometric progression: A_n = 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 Assets	Equipment book values diminishing at a constant annual depreciation percentage d follow a geometric decay curve: V_t = V0(1 – d)^t.	Equipmentbookvaluesdiminishing
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Curriculum

Mathematics

Unit 1

Unit 2

Unit 3

Unit 4

Unit 5

Unit 6

Unit 7

Lesson 1 – Arithmetic and Geometric Sequences and Series