TOPIC 5 : ARITHMETIC AND GEOMETRIC PROGRESSION

INTRODUCTION: 

 What is a sequence? It is a set of numbers which are written in some particular order.

example, take the numbers 1, 3, 5, 7, 9, . . . . 

Here, we seem to have a rule. We have a sequence of odd numbers. To put this another way, we start with the number 1, which is an odd number, and then each successive number is obtained by adding 2 to give the next odd number.

An arithmetic sequence is a sequence with the difference between two consecutive terms constant.  The difference is called the common difference. 

A geometric sequence is a sequence with the ratio between two consecutive terms constant.  This ratio is called the common ratio. 

↬ FORMULA ↫

Arithmetic formula:       t_n  =  t₁  +  (n - 1)d
t_n is the nth term, t₁ is the first term, and d is the common difference.

Geometric formula:          t_n = t₁ ^. r^{(n - 1)}
t_n is the nth term, t₁ is the first term, and r is the common ratio.

Examples

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:

b) 2/9, 2/3, 2, 6, 18,...

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

The ratio is always 3, so r = 3. Then the sixth term is (18)(3) = 54 and the seventh term is (54)(3) = 162.

c) Find the tenth term and the n-th term of the following sequence:

1/2, 1, 2, 4, 8,...

The differences don't match: 2 – 1 = 1, but 4 – 2 = 2. So this isn't an arithmetic sequence. On the other hand, the ratios are the same: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2. So this is a geometric sequence with common ratio r = 2 and a = 1/2. To find the tenth and n-th terms, we can just plug into the formula a_n = ar^{(n – 1)}:

a_n = (1/2) 2^n–1 
a₁₀ = (1/2) 2^10–1 = (1/2) 2⁹ = (1/2)(512) = 256

QUESTIONS ⇓

1. Write down the first five terms of the AP with first term 8 and common difference 7.
2. Find the 17th term of the arithmetic progression with first term 5 and common difference 2.
3. What is the common difference of the AP 11, −1, −13, −25, . . . ?
4. Write down the first five terms of the geometric progression which has first term 1 and common ratio 1 2 .
5.  Find the 10th and 20th terms of the GP with first term 3 and common ratio 2.
6. Find the 7th term of the GP 2, −6, 18, . . .,