Unit 12 – Sequences & Series: AP & GP : Basics of Statistics and Mathematics

Vocabulary flashcards covering key terms, formulas, and properties related to sequences, arithmetic progressions (AP), and geometric progressions (GP).

Sequence

An ordered list of numbers in which each term is defined by a specific rule relating it to its position n.

Term / Element

An individual member (aₙ) of a sequence, identified by its position n (the nth term).

General Term (aₙ)

A formula that expresses the nth term of a sequence as a function of n.

Finite Sequence

A sequence that contains a limited (countable) number of terms.

Infinite Sequence

A sequence with an un-ending number of terms.

Series

The sum of the terms of a sequence (a₁ + a₂ + …).

Finite Series

A series containing a finite number of addends.

Infinite Series

A series whose terms continue without end.

Progression

A sequence of numbers arranged in a specific order according to a mathematical rule (e.g., AP or GP).

Arithmetic Progression (AP)

A sequence in which the difference between consecutive terms is constant (the common difference d).

Common Difference (d)

The fixed amount added (positive) or subtracted (negative) to get successive terms in an AP.

Ascending AP

An arithmetic progression with d > 0 (terms increase).

Descending AP

An arithmetic progression with d < 0 (terms decrease).

Nth Term of an AP

Tₙ = a + (n − 1)d, where a is the first term and d the common difference.

Sum of First n Terms of an AP

Sₙ = n ⁄ 2 [2a + (n − 1)d] or Sₙ = n ⁄ 2 (first term + last term).

Arithmetic Mean (A.M.)

The average of a set of numbers; for an AP, A.M. = Sₙ ⁄ n = (first term + last term) ⁄ 2.

Geometric Progression (GP)

A sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio r.

Common Ratio (r)

The fixed number each term of a GP is multiplied by to obtain the next term (r = a₂ ⁄ a₁).

Ascending GP

A geometric progression with r > 1 (terms increase).

Descending GP

A geometric progression with 0 < r < 1 (terms decrease).

Nth Term of a GP

Tₙ = a · r^(n − 1), where a is the first term and r the common ratio.

Sum of n-Term GP (r ≠ 1, r > 1)

Sₙ = a (rⁿ − 1) ⁄ (r − 1).

Sum of n-Term GP (0 < r < 1)

Sₙ = a (1 − rⁿ) ⁄ (1 − r).

Sum of Infinite GP

S∞ = a ⁄ (1 − r), valid when |r| < 1.

Geometric Mean (G.M.)

For n numbers, the nth root of their product; for two numbers x and y, G.M. = √(xy).

A.M.–G.M. Inequality

For any positive numbers, G.M. ≤ A.M.; equality holds only when all numbers are equal.

Standard AP Term Representation

For problem-solving: three terms as a − d, a, a + d; four terms as a − 3d, a − d, a + d, a + 3d, etc.

Standard GP Term Representation

For problem-solving: three terms as a ⁄ r, a, ar; four terms as a ⁄ r³, a ⁄ r, ar, ar³, etc.

First Term (a)

The initial term from which an AP or GP starts.