Arithmetic Progression Formulas

Arithmetic Progression

Arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant difference is known as the common difference and is denoted by ‘d’. An arithmetic progression can be written as:

a, a + d, a + 2d, a + 3d, …, a + (n-1)d

Where ‘a’ represents the first term, and ‘n’ represents the number of terms in the sequence.

nth term of Arithmetic Progression

The general formula to find the nth term of an arithmetic progression is:

$a_n = a + (n-1)d$

Where:

$a_n$ represents the nth term
a is the first term
n is the number of terms
d is the common difference

Sum of the first n terms of an arithmetic progression

$S_n = \frac {n}{2} [2a + (n-1)d]$

Where:

Sn represents the sum of the first n terms
a is the first term
n is the number of terms
d is the common difference

This can also be wriiten as

$S_n=\frac {n}{2} [2a + (n-1)d]= \frac {n}{2} [a +a + (n-1)d]= \frac {n}{2} [a + a_n]$

Arithmetic Mean of two Numbers

$A= \frac {a +b}{2}$

and a , A, c are in AP

How to add n terms between two number such that they are in A.P

Suppose we want to add n terms between A and B so that result sequence is in AP
Let A₁, A₂ , A₃ , A₄ , A₅ …. A_n be the terms added between a and b
Then
b= a+ [(n+2) -1]d
or $d=\frac {b-a}{n+1}$
So terms will be
$a+ \frac {b-a}{n+1}$, $a+2\frac {b-a}{n+1}$,……., $a+n\frac {b-a}{n+1}$

How to Solve arithmetic progression Questions

If you have to take three consecutive terms in AP, Always take these

a-d , a , a + d

Here common difference is d

Benefits
When you do sum, d is eliminated, you get the value of a

If you have to take Four consecutive terms in AP, Always take these

a -3d, a -d , a + d , a+ 3d

Here common difference is 2d

Benefits
When you do sum, d is eliminated, you get the value of a

Examples and Solutions

Question 1
Find the 12th term of the arithmetic progression 5, 8, 11, 14, …

Solution: Using the formula $a_n = a + (n-1)d$, we can determine the 12th term.

Given:

a = 5 (first term

d = 8 – 5 = 3 (common difference)
n = 12 (number of terms)

Now, substitute the values into the formula:

$a_{12} = 5 + (12 – 1) \times 3 $
$a_{12} = 5 + 11 \times 3= 38$

So, the 12th term of the arithmetic progression is 38.

Question 2
Find the sum of the first 10 terms of the arithmetic progression 3, 7, 11, 15, …

Solution: Using the formula $S_n = \frac {n}{2} [2a + (n-1)d]$, we can determine the sum of the first 10 terms.

Given:

a = 3 (first term)
d = 7 – 3 = 4 (common difference)
n = 10 (number of terms)

Now, substitute the values into the formula:

$S_{10} = 10/2 [2 \times 3 + (10 – 1) \times 4] =210$

So, the sum of the first 10 terms of the arithmetic progression is 210.

Frequently Asked Questions

What is the common difference in an arithmetic progression?

The common difference is the constant difference between consecutive terms in an arithmetic progression. It is denoted by ‘d’ and can be found
by subtracting the first term from the second term or any consecutive pair of terms.

Can the common difference be negative?

Yes, the common difference can be negative. When the common difference is negative, the terms in the arithmetic progression decrease as the sequence progresses. Such an arithmetic progression is called a decreasing arithmetic progression.

Can the first term and the common difference be fractions or decimals?

Yes, the first term (a) and the common difference (d) can be fractions or decimals. The arithmetic progression formula works the same way for fractions and decimals as it does for integers. The only difference is that calculations involving fractions and decimals may require additional steps or a calculator for ease.

I hope these Arithmetic Progression Formulas helps you.

Related Articles