Algebra 3

Sequence

A number pattern in a definite order following a certain rule.

Arithmetic Sequence

What sequence is 1, 3, 5, 7, 9, …?

Geometric Sequence

What sequence is 2, 4, 8, 16, 32, …?

Triangular Number Sequence

What sequence is 1, 3, 6, 10, 15, …?

Square Number Sequence

What sequence is 0, 1, 4, 9, 16, …?

Cube Number Sequence

What sequence is 1, 8, 27, 64, 125, …?

Fibonacci Number Sequence

What sequence is 0, 1, 1, 2, 3, …?

Harmonic Sequence

What sequence is 1, 1/3, 1/5, 1/7, 1/9, …?

Progression

A sequence of numbers or quantities (called terms) in which there is always the same relation between each quantity and the one succeeding it.

1. Arithmetic Progression

2. Geometric Progression

3. Harmonic Progression

Type of Progression (3)

Arithmetic Progression

A type of progression in which each term (except the first) is obtained from the previous by adding a constant.

• a_n = a₁ + (n - 1) d

• a_n = a_m + (n - m) d

AP: nth term formula (2)

d = a₂ - a₁ = a₃ - a₂

AP: Common Difference formula

• S = n/2 (a₁ + a₂)

• S = n/2 [2a₁ + (n - 1) d]

Sum or Series of AP formula (2)

Geometric Progression

A type of progression in which, after is obtained by multiplying the preceding term by a fixed number.

• a_n = a₁r^n-1

• a_n = a_mr^n-m

nth term of GP formula (2)

S = [a₁ (1 - rⁿ)]/(1-r)

Sum or Series of GP formula

r = a₂/a₁ = a₃/a₂

Common Ratio formula

Infinite Geometric Progression

A special case of geometric progression in which there are infinite number of terms involved.

S = a₁/(1-r)

Sum or Series of IGP, when l r l < 1

S = ∞

Sum or Series of IGP, when l r l > 1

Harmonic Progression

A type of progression in which the reciprocals form an Arithmetic Progression.

Permutation

An ordered arrangement of any element of a set.

Combination

A grouping arrangement of any element of a set.

Sequence

A number pattern in a definite order following a certain rule.

Progression

The sequence of numbers called terms, each of which, after the first is derived from the preceding one.

1. Arithmetic

2. Geometric

3. Infinite

4. Harmonic

4 Types of Progression

Arithmetic Progression

A sequence of numbers called terms, each of which, after the first, is derived from the preceding one by adding a fixed number called the common difference.

Geometric Progression

A sequence of numbers called terms, each of which, after the first, is obtained by multiplying the preceding term by a fixed number called the common ratio.

Infinite Geometric Progression

A geometric progression that approaches infinity.k

Harmonic Progression

A sequence of numbers called terms in which the reciprocals from an Arithmetic Progression.

Diophantine Equations

When the number of equations is less than the number of unknowns then the equations are called as _______.

Theory of Sets

Any defined collection of elements, class, things, or objects.

• Union

• Intersection

• Difference

• Complement

4 Basic Set Operation (Theory of Sets)

Union

The ____ of sets “A” and “B” is the set of element which belongs to A or to B or to both.

Intersection

The ______ of sets A and B is the set of elements which belongs to A and also belongs to B.

Difference

The _____ of two sets A and B is the set of elements which belong to A but which do not belong to B.

Complement

The _____ of a set A is the set of elements which do not belong to A, that is; the difference between universal set and A.

Primary or Set or Logic Diagram

Venn diagram is also called? (3)

Venn Diagram

A diagram that shows all possible logical relations between a finite collection of different sets.

John Venn

Who developed Venn Diagram?

1880

When was Venn Diagram developed?

Significant Figures (Digits)

The _____ of a number are those digits that carry meaning contributing to its precision.