Monday, 30 January 2017

Topic 6: Arithmetic prigression and Geometric progression & Exercise


ARITHMETIC PROGRESSION

Definition of Arithmetic progression

Definition 1: 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.

Definition 2: The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP (arithmetic progression).

An AP can contain positive as well as negative terms. Also, the common difference can be positive, negative or zero. The nth  term of an AP is referred to be the general term of an AP. The first term, common difference and general term of an AP are denoted by a, d and an  respectively.

So, the general form of an infinite AP (containing infinite terms) can be represented as
a, a + d , a + 2d , a + 3d,…



Formula to find n term for AP 

Tn = a + (n+1) d
n = term that are given in the question
a = first term or first number
d = the common different (2nd term - 1st term) 

Example 1:
 Calculate the 7 term of the following arithmetic progression:
2, -1, -4, -6..........

Solution:
Tn = a + (n-1) d
T7 = 2 + (7 - 1) -3
T7 = 2 + (6) -3
T7 = 2 + 3
T7 = 5 


Example 2:
Calculate the 20 term of the following arithmetic progression
7, 4, 1, 2..........

Solution:
Tn = a + (n-1)d
T20 = 7 + (20-1) -3
T20 = 7 + (19) -3
T20 = 7 + (16)
T20 = 23 


Example 3:
Calculate the 5 term of the following arithmetic progression
3, 6, 9, 12..........

Solution:
Tn = a+ (n-1)d
T5 = 3 + (5-1) 3
T5 = 3+ (4) 3
T5 = 3 + 12
T5 = 15


Sum of arithmetic progression

Below is the formula to find the sum of arithmetic progression
Sn = n/2 [2a + (n-1)d] 

Example 1:
find the sum of the first 10terms of sequence below:
3,5,7,9..........

Solution:
Sn = n /2 [2a+(n-1)d]
S10 = 10 / 2 [2 (3) + (10-1) 2]
S10 = 5 [6 + (9) 2]
S10 = 5 (24)
S10 = 120








Example 2:
Find the sum of the first 45 term of the sequence below:
1, 3, 5, 7, 9..........

Solution:
Sn = n / 2 [2a + (n-1) d]
S45 = 45 / 2 [2(1) + (45-1) 2]
S45 = 22.5 [2 + (44) 2]
S45 = 22.5 [2+88]
S45 = 22.5 [90]
S45 = 2025   
 

Example 3:
Find the sum of the first 23 term of arithmetic progression
4, -3, -10..........

Solution:
Sn = n/2 [2a+(n-1)d]
S23 = 23/2 [2(4) + (23-1) -7]
S23 = 11.5 [8 + (22) -7]
S23 =11.5 [8 + -154]
S23 = 11.5 [-146]
S23 = -1679


This video teach us how to solve the arithmetic progression.
 

While this is video how to find the sum of arithmetic progression

GEOMETRIC PROGRESSION

Definition of 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. 

In a geometric progression, the ratio of any two adjacent numbers is the same. An example is 5, 25, 125, 625, ... , where each number is multiplied by 5 to obtain the following number, and the ratio of any number to the next number is always 1 to 5. Compare arithmetic progression

Formula to find n term of geometric progression

Tn = ar^n-1

Formula to find ratio

2nd term / 1st term 

Example1:
Find the 10 term?
1, -2, 4, 8..........

First find the ratio (use the formula that are given above)

r = 2ns term / 1st term
r = -2 / 1
r = -2

Second find the n term

solution:
Tn = ar^n-1
T10 = 1(-2)^10-1
T10 = -2 ^9
T10 = -512


Example 2:
Find the 10 term?
2, 6, 18, 24..........

Solution:
r = 2nd term / 1st term
r = 6 / 2
r = 3

find the n term

T10 = 2 x 3^ 10-1
T10 = 6^9
T10 = 10077696






Example 3:
Find the 5 term?
3, 6, 9, 12..........

Solution:

r = 2nd term / 1st term
r = 6 / 3
r = 2

Second find the n term

T5 = 3 (2) ^ 5 -1 
T5 = 6 ^ 4
T5 = 1296

Sum of geometric progression
The formula is 
Sn = a (1-r^n) / 1 - r)




Below there are 3 example given.

Example 1; Find the sum of the first 7 term of Geometric progression?
1, -2, 4, 8..........

Solution:
r = 2nd term / 1st term
r = -2 / 1
r = -2

Second  use the formula of sum of geometric progression

Sn = a(1-r ^ n) / 1- r)
S7 = 1 (1-(-2) ^ 7) / 1 - (-2)
S7 = 1 (129 / 3)
S7 = 43 


Example 2: Find the first 20 term of the geometric progression with first term 3 and common ratio 1.5

Solution:

Sn = a(1 - r ^ n)  / 1-r)
S20 = 3 (1 - 1.5^20) / 1 -1.5)
S20 = 3 (-3324.25673) / - 0.5)
S20 = 19945.54038


Example 3: Find the sum of the 10 term of geometric progression with the first term 5 and common ratio 2.

Solution:

Sn = a (1-r^n) / 1 - r) 
S10 = 5 (1 -2^10) / 1 - 2
S10 = 5(-1023) / -1)
S10 = -5117


Below is the example to solve the geometric progression



While this is the video about how to solve the sum of geometric progression






Exercise

1) Find the 8 term of geometric progression
1, 2, -2, 8..........

2) Calculate the 7 term of the following arithmetic progression
4, 7, -1, 8..........

3) Find the sum of the first 10 term of sequence below:
2, 5, 7, 9..........