Sequences, Series and Progressions
- A sequence is a finite or infinite list of numbers following a specific pattern. For example, 1, 2, 3, 4, 5,… is the sequence, an infinite sequence of natural numbers.
- A series is the sum of the elements in the corresponding sequence. For example, 1+2+3+4+5….is the series of natural numbers. Each number in a sequence or a series is called a term.
- A progression is a sequence in which the general term can be can be expressed using a mathematical formula.
For any finite sequence, it is generally represented as a1, a2, a3, ……an, where 1, 2, 3, …, n represents the position of the term. As the series is represented as the sum of sequences, it is represented as a1 + a2 + a3 + …. + an.
For any infinite sequence, it is generally represented as a1, a2, a3, a4, … and the infinite series is represented as a1 + a2 + a3 + ….
Arithmetic Progression
An arithmetic progression (AP) is a progression in which the difference between two consecutive terms is constant.
In arithmetic progression, the first term is represented by the letter “a”, last term is represented by “l”, the common difference between two terms is represented by “d” and the number of terms is represented by the letter “n”.
Thus, the standard form of the arithmetic progression is given by the formula,
a, a + d, a + 2d, a + 3d, a + 4d, ….
Now, consider the infinite arithmetic progression 2, 5, 8, 11, 14….
Here, first term, a = 2
Common difference = 3
Here, the common difference is calculated as follows:
Second term – first term = 5 – 2 = 3
Third term – second term = 8 – 5 = 3
Fourth term – third term = 11 – 8 = 3
Fifth term – fourth term = 14 – 11 = 3
Since the difference between two consecutive terms is constant (i.e., 3), the given progression is an arithmetic progression.
Common Difference
The difference between two consecutive terms in an AP, (which is constant) is the “common difference“(d) of an A.P. In the progression: 2, 5, 8, 11, 14 …the common difference is 3.
As it is the difference between any two consecutive terms, for any A.P, if the common difference is:
- positive, the AP is increasing.
- zero, the AP is constant.
- negative, the A.P is decreasing.
The formula to find the common difference between the two terms is given as:
Common difference, d = (an – an-1)
Where,
an represents the nth term of a sequence
an-1 represents the previous term. i.e., (n-1)th term of a sequence.
Finite and Infinite AP
- A finite AP is an A.P in which the number of terms is finite. For example the A.P: 2, 5, 8……32, 35, 38
- An infinite A.P is an A.P in which the number of terms is infinite. For example: 2, 5, 8, 11…..
A finite A.P will have the last term, whereas an infinite A.P won’t.
General Term of AP
In Arithmetic progression, an is called the general term, where n represents the position of the term in the given sequence.
The nth term of an AP
The nth term of an A.P is given by Tn= a+(n−1)d, where a is the first term, d is a common difference and n is the number of terms.
Finding nth term:
Determine the tenth term of the arithmetic progression 2, 7, 12, ….
Solution:
Given Arithmetic sequence: 2, 7, 12, …
Here, first term, a = 2
Common difference, d = 5
I.e., 7 – 2 = 5 and 12 – 7 = 5.
And now, we have to find the 10th term of AP.
Hence, n = 10
Thus, the formula to find the nth term of AP is an = a + (n-1)d
Now, substituting the values in the formula, we get
a10 = 2 + (10 – 1)5
a10 = 2 + 9(5)
a10 = 2 + 45
a10 = 47.
Therefore, 10th term of the given arithmetic sequence 2, 7, 12, … is 47.
The general form of an AP
The general form of an A.P is: (a, a+d,a+2d,a+3d……) where a is the first term and d is a common difference. Here, d=0, OR d>0, OR d<0
Sum of Terms in an AP
The formula for the sum to n terms of an AP
The sum to n terms of an A.P is given by:
Sn= n/2(2a+(n−1)d)
Where a is the first term, d is the common difference and n is the number of terms.
The sum of n terms of an A.P is also given by
Sn= n/2(a+l)
Where a is the first term, l is the last term of the A.P. and n is the number of terms.
Finding Sum of n terms of an AP:
Determine the sum of the first 22 terms of the Arithmetic Progression 8, 3, -2, ….
Here, the given arithmetic progression is 8, 3, -2, …
So, the first term, a = 8
Common difference, d = -5
I.e.,
3 – 8 = -5
-2 – 3 = -5
And, n = 22.
Now, substitute all these values in the formula: S = (n/2)[2a+(n-1)d]
S = (22/2)[2(8) + (22-1)(-5)]
S = 11 [16 + (21)(-5)]
S = 11[16 – 105]
S = 11[-89]
S = -979
Therefore, the sum of the first 22 terms of the given AP is -979.
Arithmetic Mean (A.M)
The Arithmetic Mean is the simple average of a given set of numbers. The arithmetic mean of a set of numbers is given by:
A.M= Sum of terms/Number of terms
The arithmetic mean is defined for any set of numbers. The numbers need not necessarily be in an A.P.
For example, of x, y and z are in Arithmetic progression, then y = (x + z)/2, and we can say that y is the arithmetic mean of x and z.
Basic Adding Patterns in an AP
The sum of two terms that are equidistant from either end of an AP is constant.
For example: in an A.P: 2,5,8,11,14,17…
T1+T6=2+17=19
T2+T5=5+14=19 and so on….
Algebraically, this can be represented as
Tr+T(n−r)+1=constant
Sum of first n natural numbers
The sum of first n natural numbers is given by:
Sn=n(n+1)/2
This formula is derived by treating the sequence of natural numbers as an A.P where the first term (a) = 1 and the common difference (d) = 1.
Finding Sum of first n natural numbers:
For example, if we want to find the sum of the first 10 natural numbers, we can find it as follows:
Here, n = 10.
Now, substitute the value in the formula,
Sn=n(n+1)/2
S10 = [10 (10+1)]/2
S10 = [10(11)]/2
S10 = 110/2
S10 = 55.
All the formulas related to Arithmetic Progression class 10 are tabulated below:
| First term | a |
| Common difference | d |
| General form of AP | a, a + d, a + 2d, a + 3d,…. |
| nth term | an = a + (n – 1)d |
| Sum of first n terms | Sn = (n/2) [2a + (n – 1)d] |
| Sum of all terms of AP | S = (n/2)(a + l) n = Number of terms l = Last term |
Practice Questions
- Find the sum: 34 + 32 + 30 + . . . + 10
- How many terms of the AP: 9, 17, 25, . . . must be taken to give a sum of 636?
- Find the sum of the odd numbers between 0 and 50.
- In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees that each section of each class will plant will be the same as the class in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
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