Formulas of Arithmetic Progression

A different name for this is the arithmetic sequence. 

For instance, the sequence of natural numbers 1, 2, 3, 4, 5, 6,… is an example of an arithmetic progression.

 This type of progression has a difference between any two consecutive terms (for example, 1 and 2) that is equal to 1*(2 -1). 

Even when considering even numbers and odd numbers, we can observe that the difference between any two consecutive terms will always be equal to 2. 

This is true even when considering odd numbers.

If we pay attention to the world around us, we will notice that arithmetic progression appears fairly frequently. 

Take, for instance, the number of students on a class roll, the number of days in a week, or the number of months in a year. 

Mathematicians have generalised this pattern of series and sequences into something called progressions.

Formulas for the Arithmetic Progression

When we study arithmetic progression, we will encounter two significant formulas, both of which are related to the following topics:

This is the nth term of AP.

The total of the initial n terms

Let’s learn both of these formulas here, along with some examples.

The nth term of an AP

The following is the formula that can be used to find the nth term of an AP:

an = a + (n − 1) × d

Where,

a equals the first word

d = Common difference

n equals the total number of terms

an word with the index n

Find the nth term of the sequence AP: 1, 2, 3, 4, 5…., an, if there are 15 terms total.

Given the situation, the solution is AP: 1, 2, 3, 4, 5,…, an

n=15

According to the formula that we are familiar with, an = a+(n-1d).

In the beginning, an equals 1

Common difference, d=2-1 =1

Therefore, an equals a15, which is 1+. (15-1)

1 = 1+14 = 15

Please take into consideration that the operation of the sequence is determined by the magnitude of the common difference.

In the event that “d” has a positive value, the sum of the member terms will continue 

to increase until they reach positive infinity.

In the event that “d” has a value that is negative, the member terms will continue to increase into negative infinity.

Sum of nth term of an AP

If the first term, the common difference, and the total number of terms for an AP are known, then it is possible to determine the sum of the first n terms of the AP. 

The following is an explanation of the formula for the arithmetic progression sum:

Take into consideration an AP with n different terms.

Sn = n/2[2a + (n − 1) × d]

The following is the formula for the AP sum, which may be used to compute the sum of n terms in a series.

Consider the following example of adding natural numbers up to a total of 15:

AP = 1, 2, 3,…,13, 14, 15

Given that an equals 1, d equals 2-1 = 1, and an equals 15,

Now, according to the formula, we are aware that

Sn = n/2[2a + (n − 1) × d]

S15 = 15/2[2.1+ (15-1).1] 

= 15/2[2+14] = 15/2 [16] = 15 x 8 = 120

As a result, the total of the first 15 natural numbers is equal to 120.

The tabular format, which is common in AP, is used to present the list of formulas. 

These formulas can be helpful when attempting to answer problems that are based on the idea of series and sequence.

General Form of AP a, a + d, a + 2d, a + 3d, . . .

The nth term of the AP equation a = a + (n – 1) d is as follows:

Sum of n terms in AP S = n/2[2a + (n − 1) × d]

The total of all the terms in an infinite AP, with the final term denoted by ‘l,’ is equal to n/2(a + l).

Properties of AP

• Variables are either added or subtracted from the constant.

Through the addition of a fixed amount:

If we take each term in the arithmetic progression “a, a + d, a + 2d, a + 3d, a + 4d,…………,” and add the constant quantity “k” to it, we get the following:

{a + k, a + d + k, a + 2d + k, a + 3d + k, a + 4d + k, ………..} …………. (i)

The preceding sequence begins with the term (a + k), which is the first term.

The above sequence I has a common difference that may be written as (a + d + k) – (a + k) = d.

Consequently, the terms in the aforementioned sequence I form what is known as an arithmetic progression.

If we take each term in the arithmetic progression “a, a + d, a + 2d, a + 3d, a + 4d,…” and subtract a constant quantity called “k,” we arrive at the following result:

{a – k, a + d – k, a + 2d – k, a + 3d – k, a + 4d – k, ………..} …………….. (ii)

The first term in the previously 

mentioned sequence (ii) is (a – k).

• Quadratic sequence in AP

A sequence of numbers is referred to as a quadratic sequence if the second difference between any two successive terms remains the same throughout the sequence. 

Take a look at the following illustration: 1;2;4;7;11;… It has come to our attention that each of the second discrepancies is equal to 1. 

A quadratic sequence can be defined as any sequence that possesses a common second difference.

The above sequence (ii) has a common difference that may be written as (a + d – k) – (a – k) = d.

As a consequence of this, the terms of the previously mentioned sequence (ii) form an arithmetic progression.

• An AP has equal distance from the start and the finish

The sum of the terms that are evenly spaced between the beginning and the conclusion of an AP is always the same and is equivalent to the sum of the terms that come before and after the AP.

Conclusion

A sequence of numbers that is defined as an arithmetic sequence or progression is one in which the second number in every pair of successive terms can be derived by adding a constant number to the first number in the sequence.

The common difference of an AP is the unchanging number that must be added to any given term of the AP in order to get to the following term in the AP.

 Let’s have a look at the numbers one through sixteen now: 1, 4, 7, 10, 13, 16,…

It is thought of as an arithmetic series (progression) with a difference of 3 as the common denominator.