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.

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.