Progression - Algebra

* Progression
- Also called Sequence
- These are the collection of numbers that are arranged in a fixed and orderly manner.

* Elements
- also called Terms
- it is a term used to describe the numbers arranged in a given sequence.

Two Classification of a Sequence:

1. Infinite Sequence 

- sequence whose elements are not in fixed number.
- has no ending or beginning.
e.g.
0, 1, 2, 3 . . .

. . . -2, -1, 0 
- followed by or contains three dots called ellipsis. Meaning, "and so on."

2. Finite Sequence

- Has exact number of elements or terms
- Has an ending and beginning

e.g. 1, 2, 3, 4, 5

* Definitions of Terms

1. Arithmetic Series
            - sum of the terms in an arithmetic sequence.

2. Geometric Series
            - sum of the terms in a geometric sequence.

3. Harmonic Series
            - sum of the terms in a harmonic sequence.

4. Convergent Series
            - infinite series that has a finite sum.

5. Divergent Series

            - infinite series that has no sum.

6. Alternating Series
            - the terms are arranged alternately.

* Arithmetic Sequence

            - It has a common difference between the terms.

Formula:

a_n = a₁ + (n - 1)d

Where: 

a_n = nth term 

a₁ = first term

n = last number (nth term)

d = common difference

d = preceding term - preceded term

e.g. In arithmetic sequence: 1,5,9

d = 5 - 1

   = 9 - 5

   = 4

Sum of All Terms:

S = (n/2) x [2a₁ + (n - 1)d]

Where:

S = Sum of All Terms
n = last number (nth term)
a₁ = first term
d = common difference

Example:

* Find the 27th term of the arithmetic sequence: 1, 5, 9

Given:

a₁ = 1
a₂ = 5
a₃ = 9
n = 27

Required: a₂₇

Solution:

d = 5 - 1 = 4

a₂₇ = a1 + (n - 1)d

       = 1 + (27 - 1)4

       = 1 + 104
 

a₂₇ = 105

* Geometric Sequence

            - It has a common ratio.

Formula:

a_n = a₁r^n-1

Where:

a_n = nth term
a₁ = first term
r = common ratio
n = last number (nth term)

Sum of All Terms:

If r > 1

or

If r < 1

Where:

S = sum of all terms
a₁ = first term
r = common ratio

n = last number (nth term)

e.g. In Geometric sequence: 2, 6, 18

Example:

* If the first term of a geometric progression is 3 while its common ratio is 2. Find the 5th term.

Given:

a₁ = 3

r = 2

n = 5

Required: a₅

Solution:

a₅ = a₁r^n-1

= (3)(2)^5-1

= (3)(2)⁴

= (3)(16)

a₅ = 48

* Harmonic Progression

- sequence whose elements are the reciprocal of the elements of an arithmetic sequence respectively.
- uses the formula of the arithmetic sequence.

So,

 
h = 1/a and vice versa

Example:

Find the 7th term of the harmonic progression of : ½ , ⅕, ⅛

Given:

h₁ = ½

h₂ = ⅕

h₃ = ⅛ 

Required: h₇

Solution:

* Convert first the harmonic terms to arithmetic terms. 

h₁ = 1/a₁

a₁ = 1/h₁

a₁ = 1/(½)

a₁ = 2

a₂ = 1/(⅕)

a₂ = 5

a₃ = 1/(⅛)

a₃ = 8

by using the arithmetic formula,

n = 7 

d = 5 - 2 = 8 - 5 = 3

a7 = a1 + (n - 1)d

= 2 + (7 - 1)3

= 2 + 18

a₇ = 20

* Convert a₇ to h₇

h₇ = 1 / a₇

h₇ = 1/20