nth term of an ap

In order to form a sequence of numbers, some numbers are placed in a specific order and in accordance with a specific rule. The number occurring at the nth position is referred to as the nth term of an AP, and it is denoted by the letters Tn or an. When each term in a sequence differs from its preceding term by a constant, the sequence is referred to as an arithmetic progression (abbreviated as AP). The common difference of the AP is the name given to this constant. If the first term of an arithmetic progression is a and the common difference between the first and second terms is d, then the nth term can be calculated. Consequently, in this article, we will look at the nth term of an arithmetic progression and what it means.

Progression

If the terms of a sequence are written in a specific way under specific conditions, the sequence is referred to as a progression. Generally speaking, there are three types of progression: There are three types of progression: arithmetic progression, geometric progression, and harmonic progression.

In this section, we will look at a specific type of progression known as Arithmetic Progression (AP) or Arithmetic Sequence, as well as the nth term of a given AP.

Arithmetic progression

Take a look at the following sequences.

1. 1,3,5,7,9,………..
2. 4, 8, 16, 24, ……..
3. 10, 20, 30, 40,……

Every term in the preceding sequences, with the exception of the first, is obtained by adding a fixed number (either positive or negative) to the term before it. To illustrate this, consider the sequence given in I where each term is obtained by adding 2 to the preceding term. Each term in the sequence provided in (ii) is four times greater than the preceding term, whereas in the sequence provided in (iii), each term is obtained by adding ten to the preceding term .

All of these sequences are referred to as arithmetic sequences or arithmetic progressions, which are abbreviated as AP in the AP notation.

If there is a constant term d such that the sequence a1,a2,a3,…,an is called an arithmetic progression, then the sequence is called an arithmetic progression.

a2=a1+d

a3=a2+d

For example, a4=a3+d, an=an-1+d, and so on.

A finite AP and an infinite AP are the two types of AP that can be found.

Finite AP: A finite AP is an AP that has a finite number of terms and thus is named as such. The last term is found in a finite AP.

For example: 2,6,10,14,18,22,26,30.

In this case, the first term, a1=2, is used.

The common difference, denoted by the number d=6-2=10 -6=14-10=4.

n=8 is the number of terms.

an=a8=30 was the previous term.

AP with an indefinite duration: It is an infinite AP when the number of terms in the AP is not limited to a specific number. In other words, an AP that contains an infinite number of terms is referred to as an infinite AP or a limitless AP.

For example: 1, 2, 3, 4, 5, 6, 7, and so on.

The first term, a1=1 in this case.

The common difference, denoted by the letters d=2-1=3-2=4-3=1

We cannot define the number of terms because the number of terms is n is infinite and an can’t be found out 

nth Term of an AP

On the following pages, we will look at how to calculate the nth period or general term of an AP by comparing it to its first period and the common difference. The same approach will be used to solve some problems on the AP exam as well.

In this case, a represents the first term and d represents the common difference of an AP. Then the nth term, or the general term, is defined as follows:

an=a+(n–1)d

Let’s look at how this is accomplished.

Let a1, a2, a3,….an be the APs that have been provided. Then,

a1=a

⇒a1=a+(1–1)d……(i)

Given that each term of an AP is obtained by adding a common difference to the term before it, this is a reasonable assumption.

As a result, a2=a+d 

⇒a2=a+(2–1)d (ii)

Similarly, we have a3=a2+d

⇒a3=(a+d)+d

⇒a3=a+(3–1)d……(iii)

and a4=a3+d 

a4=(a+2d)+d 

a3=(a+3d) 

a3=a+(4–1)d …….(iv)

Following the pattern established by equations (i),(ii), (iii), and (iv), we discover that

an=a+(n–1)d

As a result, the general term of an AP is = First term + (term number−1) × common difference

nth term of an AP from the end

Consider the case of an AP with the first term a and the most common difference. d. If there are m terms in the AP, the result is

The nth term from the end is equal to (m–n+1).

nth term from the end =(m–n+1)th term the beginning

⇒nth term from the end =am-n+1

⇒nth term from the end =a+(m–n+1–1)d

⇒nth term from the end =a+(m–n)d

Another point to consider is that if a polynomial has last term l, the nth term from the end is the nth term of an AP that has first term l at and the common difference is -d.

As a result, the nth term from the end equals the last term + (n–1)(–d).

The nth term from the end is equal to l–(n–1)d.

Conclusion

According to the information presented in this article, an arithmetic progression is a sequence of terms in which each term differs from its preceding term by a constant. We also learned what a sequence is, what the nth term of an arithmetic progression is, and how to solve some problems that were based on the nth term of an AP.