CHAP 7_Natural Numbers and Arithmetic Geometric Progressions
CHAPTER 7 ARITHMETIC AND GEOMETRIC PROGRESSIONS
7.1 Arithmetic Progression (A.P)
7.1.1 Definition The nth term of an arithmetic progression is given by
,
where a is the first term and d the common difference. The nth term is also known as the general term, as it is a function of n.
7.1.2 The General Term (common difference)
Example 7-1
In the following arithmetic progressions
a. 2, 5, 8, 11, ...
b. 10, 8, 6, 4, ... Write (i) the first term, (ii) the common difference, (iii) (iv)
Solution i) a. 2 b. 10
ii) a. 5-2 = 3 b. 10 -8 = 2
iii) a. = 14
b. = 2
iv) a. = 59
b. = 48
Example 7-2
How many terms are there in the following arithmetic progression?
(i) 3, 7, 11, 15, ... , 79. (ii)
Solution i)
ii)
7.1.3 The sum of the First n-Terms or
Example 7-3
a. Find the sum of 20 terms of the following arithmetic progression:
(i) 1 + 2 + 3 + 4 + ... (ii) 5 + 1 + (3) + (7) + ...
Solution (i)
(ii)
b. Find the sum of the following arithmetic progression:
(i) 13 + 17 + 21 + ... + 49. (ii) 2.3 + 2.7 + 3.1 + ... + 9.9.
Solution (i)
(ii)
7.1.4 Solving Questions on Arithmetic Progression
The questions on arithmetic progression usually involve (i) a term and another term, (ii) a term and a sum, (iii) a sum and another sum,
(iv) the definition of arithmetic progression.
(v)
Example 7-4
The third term of an arithmetic progression is 14 and the sixth term is 29. Find the first term and the common difference. Find also the tenth term and the sum to the first ten terms.
Example 7-5 The sum of the 1st 6 terms is 33 and the sum of the 1st 12 terms is 174 . Find the first term and the common difference.
…. (1)
…. (2)
from …(1) from …(2)
(2) – (1) …(3)
from …(3) a = 2
7.2 Geometric Progression (G.P)
7.2.1 Definition
The nth term of a geometric progression is given by , where a is the first
7.1 Arithmetic Progression (A.P)
7.1.1 Definition The nth term of an arithmetic progression is given by
,
where a is the first term and d the common difference. The nth term is also known as the general term, as it is a function of n.
7.1.2 The General Term (common difference)
Example 7-1
In the following arithmetic progressions
a. 2, 5, 8, 11, ...
b. 10, 8, 6, 4, ... Write (i) the first term, (ii) the common difference, (iii) (iv)
Solution i) a. 2 b. 10
ii) a. 5-2 = 3 b. 10 -8 = 2
iii) a. = 14
b. = 2
iv) a. = 59
b. = 48
Example 7-2
How many terms are there in the following arithmetic progression?
(i) 3, 7, 11, 15, ... , 79. (ii)
Solution i)
ii)
7.1.3 The sum of the First n-Terms or
Example 7-3
a. Find the sum of 20 terms of the following arithmetic progression:
(i) 1 + 2 + 3 + 4 + ... (ii) 5 + 1 + (3) + (7) + ...
Solution (i)
(ii)
b. Find the sum of the following arithmetic progression:
(i) 13 + 17 + 21 + ... + 49. (ii) 2.3 + 2.7 + 3.1 + ... + 9.9.
Solution (i)
(ii)
7.1.4 Solving Questions on Arithmetic Progression
The questions on arithmetic progression usually involve (i) a term and another term, (ii) a term and a sum, (iii) a sum and another sum,
(iv) the definition of arithmetic progression.
(v)
Example 7-4
The third term of an arithmetic progression is 14 and the sixth term is 29. Find the first term and the common difference. Find also the tenth term and the sum to the first ten terms.
Example 7-5 The sum of the 1st 6 terms is 33 and the sum of the 1st 12 terms is 174 . Find the first term and the common difference.
…. (1)
…. (2)
from …(1) from …(2)
(2) – (1) …(3)
from …(3) a = 2
7.2 Geometric Progression (G.P)
7.2.1 Definition
The nth term of a geometric progression is given by , where a is the first