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
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)
Below is the formula to find the sum of arithmetic progression
Sn = n/2 [2a + (n-1)d]
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
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.
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)
Example 2: Find the first 20 term of the geometric progression with first term 3 and common ratio 1.5
Below is the example to solve the geometric progression
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..........