TOPIC8: ARITHMETIC PROGRESSION & GEOMETRIC PROGRESSION

ARITHMETIC PROGRESSION

• An arithmetic sequence or progression is defined as a sequence of numbers in which for every pair of consecutive terms, the second number is obtained by adding a fixed number to the first one.

   

Arithmetic progression examples

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 

Example1:

-  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

Example2

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

 

Example3

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

 

GEOMETRIC PROGRESSION

• A geometric progression is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by "r". The common ratio is obtained by dividing any team by preceding term.

 

 

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

Example 2: 

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

- r = 2nd term/1st term

     = -8/2

     = -3