Lesson 1 - Arithmetic and Geometric Sequences and Series

• Sequence: A function whose domain is the set of positive integers ℤ⁺ = {1, 2, 3, ...} and whose range is a subset of real numbers ℝ. Its terms are denoted T1, T2, ..., Tn.
• 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⁺.
• 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⁺.
• Arithmetic Mean: The arithmetic mean of two real numbers a and b is (a + b) / 2. If three numbers form an A.P. (a, b, c), then b = (a + c) / 2.
• 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, b, c), then b² = ac.

• 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⁺).
• 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.
• 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_∞.

• 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.

An Arithmetic Progression (A.P.) is defined by the invariant linear recurrence relation: T_(n+1) - T_n = 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: T_n = a + (n - 1)d. Notice that T_n 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.

A Geometric Progression (G.P.) is characterized by an invariant multiplicative factor: T_(n+1) / T_n = r (constant common ratio, r ≠ 0). General Term: T1 = a, T2 = ar, T3 = ar², giving: T_n = a · r^{n - 1}.

Partial Sum Formula: Multiplying the partial sum S_n = a + ar + ar² + ... + ar^n-1 by r and subtracting gives: S_n - r S_n = a - a rⁿ -> S_n = [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: lim_{n -> ∞} rⁿ = 0. Therefore, the infinite sum converges to: S_∞ = lim_{n -> ∞} S_n = 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)}.

The Greek capital letter Sigma (∑) is used to denote series sums compactly: ∑_{r=1}ⁿ T_r = T1 + T2 + ... + Tn.

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).

(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.

Index Shift Property: To evaluate ∑{r=k}ⁿ T_r where k > 1, apply: ∑{r=k}ⁿ T_r = ∑{r=1}ⁿ T_r - ∑{r=1}^k-1 T_r.

Problem Statement: Given the arithmetic progression (35, 31, 27, 23, ...): (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.

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 T_n > 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).

• 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.
• Find the sequence and calculate the sum of its first 6 terms.

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, 6, 12, 24, 48, ...). Step 6: Sum of first 6 terms: S6 = [a(r⁶ - 1)] / (r - 1) = [3(2⁶ - 1)] / (2 - 1) = 3(64 - 1) = 3 × 63 = 189.

• 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.
• 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).

• Financial Mathematics & Compound Interest: Bank investments compounding annually at interest rate i form a geometric progression: A_n = P(1 + i)ⁿ.
• Nuclear Chain Reactions: In uranium fission, each neutron generates 2 or 3 secondary neutrons, causing nuclear power output to expand geometrically.
• 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.