Learn the Arithmetic Progression Formula from our best teachers. You can find the AP, Common difference, the nth term, the sum of AP with Examples.
Arithmetic Progression Formula
Progression Types:
In mathematics, there are three different types of progressions. The name is given below:
- Arithmetic Progression (A.P.)
- Harmonic Progression (H.P.)
- Geometric Progression (GP)
Arithmetic Progression
A.P. is a sequence of numbers in order in which the difference between two consecutive numbers is a constant value. It is also called the Arithmetic Sequence. It is a sequence where all the terms are obtained by adding a fixed number to its previous term.
For example,
we have,
1, 3, 5, 9, 13, 17, 21, 25, 29, 33, …
- a = 1 (the first term)
- d = 4 (the “common difference” between terms)
In general form, an arithmetic sequence can write like this:
{a, a+d, a+2d, a+3d, … }.
Using the above example, we get the following:
{a, a+d, a+2d, a+3d, … } = {1, 1+4, 1+2×4, 1+3×4, … } = {1, 5, 9, 13, … }
If we observe in our formal lives, we come across Arithmetic progression quite often. For example, In a class, Roll the number of students’ days in a week or months in a year. This pattern of series and sequences has been generalized in Maths as progressions.
Arithmetic Progression Formula:
For the first term ‘a’ of an Arithmetic Progression and common difference ‘d’, there are some arithmetic progression formulas given below which will help us to solve various problems related to AP:
- Common difference of an AP = d = a2 – a1 = a3 – a2 = a4 – a3 = … = an – an-1
- Sum of n terms of an AP = Sn = n/2(2a+(n-1)d) = n/2(a + l), where l is the last term of the arithmetic progression.
- nth term of an AP = an = a + (n – 1)d
First Term of Arithmetic Progression
Here some formulas related to the Arithmetic Progression are given below:
As the name suggests,
An Arithmetic Progression’s first term is the progression’s first number. It can also be written in terms of common differences, as follows;
- a, a + d, a + 3d, a + 2d, a + 4d, ………. ,a + (n – 1) d
Here “a” is the first term of the progression.
Common Differences in Arithmetic Progression
If the first term = a1,
The second term = a1+d,
The third term = a1+d+d = a1+2d,
and the fourth term = a1+2d+d=, a1+3d and so on.
The formula of common difference is given below:
- d = a2 – a1 = a3 – a2 = ……. = an – an – 1
Where “d” is a common difference, it can be positive, negative, or zero.
nth Term of an AP
The formula of “nth” term of an AP is given below:
- an = a + (n − 1) × d
Where
a = First term of AP
d = Common difference of AP
n = number of terms of AP
an = nth term of AP
Example:
Find the nth term of AP: 1, 2, 3, 4, 5…., and, if the number of terms is 25.
Solution: Given that,
AP = 1, 2, 3, 4, 5…., an
n=25
By the formula we know, an = a+(n-1)d
First-term, a =1
The common difference, d=2-1 =1
Therefore, an = a25 = 1+(25-1)1 = 1+24 = 25
Note: The behavior of the sequence depends on the value of a common difference.
- When the value of “d” is negative, the member terms grow towards negative infinity.
- When the value of “d” is positive, the member terms will grow toward positive infinity.
Types of Arithmetic Progression
There are two types of Arithmetic Progression,
- Finite Arithmetic Progression
- Infinite Arithmetic Progression
1. Finite Arithmetic Progression: An Arithmetic progression containing a finite number of terms is called finite AP. A finite AP has the last term.
For example 3,5,7,9,11,13,15,17,19,21………….
2. Infinite Arithmetic Progression: An Arithmetic Progression that does not have a finite number of terms is called infinite AP. Such APs do not have the last term.
For example: 5,10,15,20,25,30, 35,40,45………………
Sum of N Terms of A.P.
In an Arithmetic progression, if the first term, common difference, and the complete terms are known, then we can calculate the sum of the first n terms.
The formula for the arithmetic progression sum explained below:
- Sn = n/2 [2.a + (n − 1) × d]
It is the A.P. sum formula to find the sum of n terms in a series.
Example:
Let us take the example of adding natural numbers up to 28 numbers.
A.P. = 1, 2, 3, 4, 5,……………., 11, 12, 13, 14, 15
Given, a = 1, d = 2-1 = 1 and an = 28
Now, by the formula, we know;
Sn = n/2 [2.a + (n − 1) × d]
S15 = 28/2[2.1+(28-1).1]
= 28/2[2+27]
= 28/2 [29]
S15 = 14 x 29
= 406
Hence, the sum of the first 28 natural numbers is 406.
Sum of A.P. when the Last Term is Given
The formula for the arithmetic progression sum explain below:
- S = n/2 (first term + last term)
It is the formula of the Sum of A.P. when the Last Term.
Arithmetic Progressions Solved Examples
Example 1:
Find the value of n if a = 15, d = 6, and an = 85.
Solution: Given that,
a = 15, d = 6, an = 85
From the formula of a general term, we have:
an = a + (n − 1) × d
⇒85 = 15 + (n − 1) × 6
⇒(n − 1) × 6 = 85 – 15 = 70
or, (n − 1) = 70/ 5
⇒(n − 1) = 35
⇒n = 35 + 1
Therefore, n = 36 ( Answer )
Example 2:
Find out the sum of the first 25 multiples of 5.
Solution:
The first 25 multiples of 5 are: 5, 10, 15, ….., 120
Here, a = 5, n = 25, d = 10 -5 =5
We know,
S25 = n/2 [2a + (n − 1) × d]
S25 = 25/2[2 (5) + (25 − 1) × 5]
or, S25 = 25/2[10 + 120]
S25 = 25/2 [130]
Therefore, S25 = 1625 ( Answer )
Some Examples Based on A. P Formula
Example 3:
What is the general term of the arithmetic progression (A.P.) -5, -(1/2), 2…
Solution:
The sequence is given -5, -(1/2), 2…
Here,
the first term is a=-5, and
the common difference d is = -(1/2) -(-5) = -(1/2)+5 = 9/2
By A.P. formulas the general term of an A.P. is calculated by the formula:
an = a+(n-1)d
an = -5 +(n-1) 9/2
= -5+ (9/2)n – 9/2
= 9n/2 – 1/2
Therefore, the general term of the given A.P. is:
Answer: an = 9n/2 – 1/2
Example 4: Which term of the A.P. 3, 8, 13, 18,… is 68?
Solution:
The given sequence is 3, 8, 13, 18,…
Here the 1st term is a=3, and the common difference d is, = 8-3= 13-8=…=5
Let us assume that the nth term is,
an=68
Substitute all these values in the general term of an arithmetic progression:
an = a+(n-1)d
68 = 3+(n-1)5
68 = 3+5n-5
or, 68 = 5n -2
70 = 5n
n = 14
Answer: ∴ 68 is the 16th term.
Example 5:
What is the sum of the 1st five (5) terms of the arithmetic progression (A.P.) whose 1st term is 3 and 5th term is 11?
Solution: We have a1 = a = 3 and a5 = 11 and n = 5.
Using the A.P. formula for the sum of n terms, we have
Sn = (n/2) (a + an)
⇒ S5 = (5/2) (3 + 11)
= (5/2) × 14
= 35
Answer: The required sum of the first 5 terms is 35.
Some Important Examples
Example 6:
What is the 20th term of the A.P. sequence or arithmetic sequence 24, 21, 18, 15, 12, … ?
Solution:
This sequence is descending, so it has a difference of -5 between each pair of numbers.
The values of d and a are:
a = 24 (the first term)
d = -5 (the “common difference”)
The rule can calculate:
xn = a + d(n-1)
= 24 + -5(n-1)
= 24 – 5n + 5
Therefore, xn= 29 – 5n
So, the 20th term is:
x20 = 29 – 5 × 20
= 29 – 100
= – 71 ( Answer)
Example 7: What is the 30-second term of the arithmetic sequence or A.P. sequence -12, -7, -2, 3, … ?
Solution: This sequence has five (5) differences between each pair of numbers.
The values of a and d are:
a = -12 (the first term)
d = 5 (the “common difference”)
The rule can calculate:
xn = a + d(n − 1)
= -12 + 5(n − 1)
= -12 + 5n − 5
Therefore, xn = 5n − 17
So, the 32nd term is:
x32 = 5 × 32 − 17
= 160 − 17
= 143 ( Answer )
Example 8:
Find out the sum of the first 30 terms of the arithmetic sequence: 55, 50, 45, 40, 35, 30 … ?
Solution: 50, 45, 40, 35, 30 …
The values of d, a, and n are:
a = 55 (the 1st term)
d = -5 (here, d is the common difference between them)
n = 30 ( here, n indicates how many terms to add up)
By using the sum of A.P. or the Arithmetic Progression formula – Sn = n/2(2a + (n – 1)d), we get:
S30 = 30/2(2 × 55 + 29 × -5))
=15(110 – 145)
= 15 × -35
Therefore, S30= -525 ( Answer )
Practice Problems on AP
You will find the below questions based on Arithmetic sequence formulas by yourself and solve them for good practice.
1: Find the an and 10th term of the progression: 3, 1, 17, 24, ……
2: If a = 2, d = 3 and n = 90. Find an and Sn.
3: An AP’s 7th term and 10th terms are 12 and 25. Find the 12th term.
4: Consider an arithmetic progression whose first term and common difference are 100. Suppose the nth term of this progression is equal to 100! Find n.
5: What is the seventh term of the arithmetic progression 2, 7, 12, 17, …?
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