Arithmetic and Geometic Progression

ARITHMETIC AND GEOMETIC PROGRESSION

Definition:

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, etc, is an arithmetic progression with common difference of 2.

Formula: Tn = a + (n-1) d

                        a = 1^st term

                        n = nth term

                        d = common difference

What is the 10th term?

T10 = 1 + (10-1) 2

        = 1 + (9) (2)

        = 1 + 18

        = 19

What is the first 3 terms?

T2 = 1 + (2-1) 3          T3 = 1 + (3-1) 3

      = 4                              = 7

What is the 16th term?

T16 = 8 (16-1) = 3

        = 8 + (15) (-3)

        = 8 + (-45)

        = -37 

Example:

-  Write down the first four terms of AP with first term 8 and difference 7.

T2 = 8 + (2-1) 7          T3 = 8 + (3-1) 7          T4 = 8 + (4-1) 7

      = 15                            =22                             = 29

- Write down the first four terms of AP with first term 2 and difference -5.

T2 = 2 + (2-1) -5         T3 = 2 + (3-1) -5         T4 = 2 + (4-1) -5

      = -3                             = -8                             = -13

- Write down the 10th and 19th terms of the AP.

i) 8, 11, 14...

T10 = 8 + (10-1) 3        T19 = 8 + (19-1) 3

        = 8 + (9) (3)                  = 8+ (18) (3)

        = 8 + 27                         = 8 + 54

        = 35                              = 62

ii) 8, 5, 2...

T10 = 8 + (10-1) -3       T19 = 8 + (19-1) -3

        = 8 + (9) (-3)                = 8 + (18) (-3)

        =8 + (-27)                      = 8 + (-54)

        = -19                             = -46

SUM OF ARITHMETIC PROGRESSION

Definition:

The sum of a finite arithmetic progression is called an arithmetic series. The behavior of the arithmetic progression depends on the common difference d. If the common difference  is: Positive, then the members (terms) will grow towards positive infinity.

Formula: Sn = n/2 [2a + (n-1)d]

                        a = 1st term

                        n = nth term

                        d = difference

                     Sn = Sum of AP

Find the sum of the first 50 terms of the AP?

S50 = 50/2 [2 x 1 + (50-1) (2)]

S50 = 2,500

Example:

1) Find the sum of the first 37 of AP 4, -3, -10...

a = 4

d = -7

S37 = 37/2 [2 x 4 + (37-1) (-7)]

S37 = -4,514

2) An AP has first term 4 and difference 1/2. Find:
- a) Sum of the first 25 terms          - b) Sum of the first 100 terms

a = 4                                                              a = 4

d = 1/2                                                          d = 1/2

n = 25                                                            n = 100

S25 = 25/2 [2 x 4 + (25-1) (1/2)]             S100 = 100/2 [2 x 4 + (100-1) (1/2)]

S25 = 250                                                  S100 = 2.875

GEOMETRIC PROGRESSION

Introduction:
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Formula: Tn = ar ^n-1

                        a = 1^st term

                        r = common ratio

                        n = nth term

2, 6, 18, 54,...     r = 2^nd term/1^st term 

                                = 6/2 

                                = 3

Find the 15th term of the GP?

T15 = 2 x 3 ^15-1

       = 9, 565,938

Example:

1) Find the 10th and 17th term of GP with first term 3 and common ratio 2.

- a) a = 3                 b)   a = 3

       r = 2                         r = 2

      n = 10^th                         n = 17th

T10 = 3 x 2 ^10-1           T10 = 3 x 2 ^17-1

        = 1,536                    = 196, 608

2) Find the 7th term of the GP 2, -6, 18....

- r = 2nd term/1st term

     = -8/2

     = -3

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