Ch 10Sequence and Series — Class 11 Mathematics, concept-first.

A sequence is an arrangement of numbers in a definite order, formed according to some definite rule. Formally, a sequence is a function whose domain is the set of natural numbers (or, for a finite seq…

A sequence is called an Arithmetic Progression (A.P.) if the difference between every pair of consecutive terms is the same constant, called the common difference, usually denoted : If the first term…

If three numbers are in A.P. (in that order), then is called the arithmetic mean (A.M.) of and . Being in A.P. means the common difference is the same on both sides: .

A sequence (with every term nonzero) is called a Geometric Progression (G.P.) if the ratio of every pair of consecutive terms is the same constant, called the common ratio, usually denoted : If the fi…

If three positive numbers are in G.P. (in that order), then is called the geometric mean (G.M.) of and . Being in G.P. means the common ratio is the same on both sides: .

Let and be two positive real numbers, with arithmetic mean and geometric mean (Sections 3 and 5). A fundamental and constantly-used fact relates the two: with equality holding if and only if .

A series is called an Arithmetic-Geometric Progression (A.G.P.) if its th term is the product of the th term of an A.P. and the th term of a G.P. If the A.P. is and the G.P. is , the A.G.P.

Section 4 derived the sum of the first terms of a G.P. as for . This section asks: does it make sense to sum infinitely many terms of a G.P., i.e.

Three particular sums recur so often — both later in this syllabus and in the two exercises above — that they are worth deriving once and for all: the sum of the first natural numbers, of their square…

This chapter built the algebra of sequences and series around two fundamental progressions and the ways they combine.