Arithmetic Progression

A sequence where a term is found by adding a constant to its previous term is called an arithmetic progression. In other words, the difference between two consecutive terms in an AP will be the same. The difference is known as the common difference (d), and the first term is denoted by a or a₁, and the n^th term is denoted by a_n. For example, 3, 6, 9, 12, 15, … is an arithmetic progression. Here, a is 3, and the constant d is also 3. In its general form, AP can be expressed as a, a + d, a + 2d,…, a + (n - 1)d.

There are mainly two types of progressions: arithmetic and geometric (GP). Here, we will be discussing the difference between AP and GP:

Key formulas for arithmetic progression for calculating common difference, the nth term, and the sum of terms. 

The common difference in AP: 
The common difference of a sequence is the difference between two consecutive terms, and it is denoted by d:
d = a2 - a = a3 - a2 = a4 - a3 = … = an - an-1

The nth term of an AP: 
The nth term of an AP can be expressed as an = a + (n - 1)d

Where n is the number of terms
a is the first term 
a2 is the second term
a3 is the third term
d is the common difference
an is the nth term of the sequence

Now we will learn how to find the sum of the first n terms of an AP. So let’s consider the first n terms of an AP as a1, a1 + d, a1 + 2d, a1 + 3d, ….., a1 + (n - 1)d, and the sum of these n terms is sn
Then the sum of nth terms:
Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ….., + (a1 + (n - 1)d) 

Now let's reverse the sequence and add corresponding terms. It can be written as:
Sn = an + (an - d) + (an - 2d) + (an - 3d) + ….., + (an + (n - 1)d) 

Adding these two equations: 
Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ….., + (a1 + (n - 1)d) 
Sn = an + (an - d) + (an - 2d) + (an - 3d) + ….., + (an + (n - 1)d) 

By adding these, we get: 2sn = (a1 + an) + (a1 + an) + (a1 + an) + …. + (a1 + an)
All the d terms are cancelled out here, so:
2sn = n(a1 + an)
Sn = n(a1 + an)/2, let's consider it as equation 1

Substituting the formula to find an in equation 1, an = a1 + (n - 1)d
Then, Sn = n/2 (a1 + a1 + (n - 1)d) 
= n/2 [(2a + (n - 1)d], let’s consider this equation as 2

Equations 1 and 2 are used to find the sum of an arithmetic progression.  
So, Sum of nth term: sn = n/2 (2a + (n - 1)d) and sn = n/2(a + l)

The nth term of an AP is used to find any term in the sequence without listing all the previous terms. There is a formula that we can use to find the nth term, which is an = a + (n - 1)d. 
For example, for the AP: 2, 4, 6, 8, 10,… we find the nth term. Here, a = 2, d = 2
an = a + (n - 1)d
= 2 + (n - 1)2
= 2 + 2n -2
an = 2n

To find the nth term of an AP, we use the formula an = a + (n - 1)d. For example, for the sequence 10, 20, 30, 40, …, here, a = 10 and d = 10. 
So, the 5th term a5 = 10 + (5 - 1)10 = 10 + 40 = 50. Similarly, to find the 55th term, we can use the formula, 
an = a + (n - 1)d
a55 = 10 + (55 - 1) 10
= 10 + 54 × 10 
= 10 + 540 = 550. 
So, the 55th term of this AP is 550.

The arithmetic sequence explicit formula is the nth term of an AP, it is used to find any term of the sequence.

Now we will learn to find the sum of arithmetic progression. The sum of the first ‘n’ terms can be calculated with the help of the formula mentioned below.

When the nth term is unknown, the sum of n is calculated using the formula:

S_n = n/2 (2a + (n - 1)d)

When the nth term is known, the sum of n is calculated using: S_n = n/2 (a + a_n)
 

Where,

n is the number of terms

a is the first term

d is the common difference

a is the first term

a_n is the nth term 

In our real world, we use AP from basic counting to calculate the interest rate. Here, we will learn some applications of arithmetic progression.

• We use AP to predict the upcoming term in a sequence. This is achieved by adding a constant difference. For e.g., we can predict the next number in the sequence 3, 7, 11,.....

• For predicting future values based on past trends, we use an AP

• To calculate the interest rates, appropriations, and loan disbursements, we use AP. 

• For seating arrangements such as arranging seats in rows and columns with equal distance, we use AP