Flash Cards: Sequences and Series

A sequence is an ordered collection of numbers defined by a specific rule, starting with a first member and continuing with subsequent members.

A finite sequence has a fixed number of terms. For example, the sequence of ancestors listed is finite with 10 terms.

The n-th term of a sequence is denoted by a_n, where n indicates the position of the term.

An arithmetic progression (AP) is a sequence in which each term after the first is obtained by adding a constant difference to the previous term.

The common difference, denoted as d, is the constant value added to each term in an AP to obtain the next term.

The n-th term of an AP can be found using the formula a_n = a + (n-1)d, where a is the first term.

The sum S_n of the first n terms in an AP is given by S_n = n/2 [2a + (n-1)d].

A geometric progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a constant known as the common ratio.

The common ratio, denoted as r, is the constant factor that each term is multiplied by to get the next term in a GP.

The n-th term of a GP is given by a_n = ar^(n-1), where a is the first term and r is the common ratio.

The sum S_n of the first n terms in a GP is given by S_n = a(1 - r^n) / (1 - r) if r ≠ 1.

The Fibonacci sequence is defined by a_1 = 1, a_2 = 1, and a_n = a_{n-1} + a_{n-2} for n > 2.

The arithmetic mean (A.M.) is always greater than or equal to the geometric mean (G.M.), i.e., A ≥ G.

The arithmetic mean of two positive numbers a and b is A = (a + b) / 2.

The geometric mean of two positive numbers a and b is G = √(ab).

Assuming that all sequences have a mathematical formula; some sequences, like the prime numbers, do not.

Sigma notation (∑) is a compact way to denote the sum of a sequence, indicating the starting and ending terms.

An explicit formula defines the n-th term directly, while a recursive formula defines terms based on previous terms.