TOPIC 6: ARITHMETIC AND GEOMETRIC PROGRESSION

 ARITHMETIC AND GEOMETRIC PROGRESSION

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 (r) is obtained by dividing any term by the preceding term.

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 

Example 1:

Find the common difference and the next term of the following sequence:

a) 3, 11, 19, 27, 35,...

To find the common difference, we have to subtract a pair of terms. It doesn't matter which pair we pick, as long as they're right next to each other:

11 – 3 = 8

19 – 11 = 8

27 – 19 = 8

35 – 27 = 8

The difference is always 8, so d = 8. Then the next term is 35 + 8 = 43.

Find the common ratio and the seventh term of the following sequence.

Example 2:

 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

Example 3:

Find a₂₀ of a geometric sequence if the first few terms of the sequence are given by
-1/2 , 1/4 , -1/8 , 1 / 16 , ...  

Solution to Example 3: 

We first use the first few terms to find the common ratio

r = a₂ / a₁ = (1/4) / (-1/2) = -1/2

r = a₃ / a₂ = (-1/8) / (1/4) = -1/2

r = a₄ / a₃ = (1/16) / (-1/8) = -1/2

The common ration r = -1/2. We now use the formula a_n = a₁ * r^n-1 for the n th term to find a₂₀ as follows.

 a₂₀ = a₁ * r^20-1 
= (-1/2) * (-1/2)^20-1 = 1 / (20²⁰)

QUESTIONS FOR ARITHMETIC AND GEOMETRIC PROGRESSION

Question 1

Write down the 8th term in the Geometric Progression 1, 3, 9, ...

Question 2

Find the sum of the following Arithmetic Progression 1,3,5,7....,199

Question 3

The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?

1. 300
2. 120
3. 150
4. 170
5. 270