Arithmetic Progression
Arithmetic Progression (AP) is a sequence of numbers in a particular order. If we observe in our regular lives, we come across progression quite often. For example, Roll numbers of a class, days a week or months in a year. Did you notice that counting numbers, even or odd numbers, all follow a particular pattern? This pattern of series and sequences has been generalized in Maths as progressions. Let us learn here AP definition, formula and solve problems or questions based on it.
In mathematics, there are three different types of progressions. They are:
- Arithmetic Progression(AP)
- Geometric Progression(GP)
- Harmonic Progression(HP)
Definition
A progression is a special type of sequence for which it is possible to obtain a formula for the nth term. The Arithmetic Progression is the most commonly used sequence in maths with easy to understand formulas. Let us see its three different types of definition.
Definition 1: It is a mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP.
Definition 2: 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 3: 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.
Now, let us consider the sequence, 1, 4, 7, 10, 13, 16,… is considered as an arithmetic sequence with common difference 3.
Important Terms of AP
In this progression, for a given series, the terms used are first term, common difference between the two terms and nth term.
Suppose, a_{1}, a_{2}, a_{3}, ……………., a_{n} is an AP, then;_{ }
the common difference “ d ” can be obtained as;
d = a_{2} – a_{1} = a_{3} – a_{2} = ……. = a_{n} – a_{n – 1} |
Where “d” is a common difference.
The AP can also be written in terms of common difference, as follows;
a, a + d, a + 2d, a + 3d, a + 4d, ………. ,a + (n – 1) d |
where “a” is the first term of the progression.
General Form of A. P
Consider an AP to be: a_{1}, a_{2}, a_{3}, ……………., a_{n}
Position of Terms | Representation of Terms | Values of Term |
---|---|---|
1 | a_{1} | a + d = a + (1-1) + d |
2 | a_{2} | a + 2d = a + (2-1) + d |
3 | a_{3} | a + 3d = a + (3-1) + d |
4 | a_{4} | a + 4d = a + (4-1) + d |
. | . | . |
. | . | . |
. | . | . |
. | . | . |
n |
a_{n} |
a + (n-1)d |
Arithmetic Progression Formula
In AP, we have the formula to find nth term, sum of nth term and sum of AP when last term is given.
The nth term formula of an AP
To find the nth term and sum of n terms of an arithmetic series, the AP formula is given as follows:
The formula for finding the n-th term of an AP is:
a_{n} = a + (n − 1) × d |
Where,
a = First term
d = Common difference
n = number of terms
a_{n} = nth term
Note:
The finite portion of an AP is known as finite AP and therefore the sum of finite AP is known as arithmetic series. The behaviour of the sequence depends on the term common difference.
- If the term “d” is positive, then the member terms will grow towards positive infinity
- If the term “d” is negative, then the member terms grow towards negative infinity
Sum of nth Terms of AP
For any progression, the sum of n terms can be easily calculated. For an AP, the sum of the first n terms can be calculated if the first term and the total terms are known. The formula for the arithmetic progression sum is explained below:
Consider an AP consisting “n” terms.
S = n/2[2a + (n − 1) × d] |
This is the AP sum formula to find the sum of n series.
Proof:
Consider an AP consisting “n” terms having the sequence a, a + d, a + 2d, ………….,a + (n – 1) × d
Sum of first n terms = a + (a + d) + (a + 2d) + ………. + [a + (n – 1) × d] ——————-(i)
Writing the terms in reverse order,
we have: S = [a + (n – 1) × d] + [a + (n – 2) × d] + [a + (n – 3) × d] + ……. (a) ———–(ii)
Adding both the equations term wise, we have
2S = [2a + (n – 1) × d] + [2a + (n – 1) × d] + [2a + (n – 1) × d] + …………. + [2a + (n – 1) ×d] (n-terms)
2S = n × [2a + (n – 1) × d]
S = n/2[2a + (n − 1) × d]
Sum of AP when Last Term is Given
Formula to find sum of AP when first and last terms are given as follows:
S = n/2 (first term + last term) |
Examples
Below are the problems to find the nth terms and sum of the sequence are solved using AP sum formulas in detail. Go through once and solve the practice problems to excel your skills.
Example 1: Find the value of n. If a = 10, d = 5, a_{n} = 95.
Solution:
Given, a = 10, d = 5, a_{n} = 95
From the formula of general term, we have:
a_{n} = a + (n − 1) × d
95 = 10 + (n − 1) × 5
(n − 1) = 17
n = 18
Example 2: Find the 20th term for the given AP:
3, 5, 7, 9, ……
Solution:
Given, a = 3, d = 5 – 3 = 2, n = 20
a_{n} = a + (n − 1) × d
a_{n} = 3 + (20 − 1) × 2
a_{n} = 3 + 38
⇒a_{n} = 41
Example 3: Find the sum of first 30 multiples of 4.
Solution: Given,
a = 4, n = 30, d = 4
We know,
S = n/2 [2a + (n − 1) × d]
S = 30/2[2 (4) + (30 − 1) × 4]
S = 15[8 + 116]
S = 1860
Problems on AP
Find the below questions based on Arithmetic sequence formulas and solve it for good practice.
Question 1: Find the a_n and 10th term of the progression: 3,10,17,24.
Question 2: If a = 2, d = 3 and n = 90. Find a_{n} and S_{n}.
Question 3: The 7th term and 10th terms of an AP is 12 and 25. Find the 12th term.