General term of the PA

O general term of a arithmetic progression (PA) is a formula used to find any term of an AP, indicated by a_no, when your firstterm (The₁), the reason (r) and the numberinterms (n) that this PA has are known.

The general term formula of progressionarithmetic is as follows:

The_no = the₁ + (n – 1)r

This formula can be obtained from an analysis of the terms gives PAN. For this, it is necessary to know some elements and characteristics of arithmetic progressions, which will be discussed briefly below.

See too:Sum of terms of an arithmetic progression

What is a PA?

One progressionarithmetic is sequence of numbers where each term (number) is the result of the sum of its predecessor with a constant, called reason. The terms of an AP are indicated by indexes, so that each index determines the position of each element of the progression. See an example:

A = (a₁, a₂, a₃, … The_no)

If the_no - a_{n - 1} = k for all n, so the above sequence is a progressionarithmetic.

See too: Geometric progression

Finding the formula of the general term of the PA

Knowing that each term of a PAN is equal to its previous one added to a constant, we can write the BP terms in function of the first term. In the progression A = (1, 3, 5, 7, 9, 11, 13, … a_no), for example, we will have:

The₁ = 1

The₂ = 1 + 2

The₃ = 1 + 2·2

The₄ = 1 + 2·3

The₅ = 1 + 2·4

The₆ = 1 + 2·5

The₇ = 1 + 2·6

…

The_no = 1 + 2·(n - 1)

This is the formula used to find any term, that is, the termgeneral of the PA given as an example.

Knowing that the_no represents any term of a PA, we can try to find the termgeneral of a progressionarithmetic whose terms are unknown. For this, consider an AP that has n terms. Know that the₁ is the first, the_no is the last and the reason is r.

We can write the terms of this PAN in function of the first as follows:

The₁ = the₁

The₂ = the₁ + r

The₃ = the₁ + r + r = a₁ + 2r

The₄ = the₁ + r + r + r = a₁ + 3r

…

The_no = the₁ + r + r + r … + r = a₁ + r (n - 1)

Thus, by rewriting the last equality and rearranging the terms of the last member, we will have:

The_no = the₁ + (n – 1)r

This is formula of termgeneral of arithmetic progression.

Example

what is the hundredth term of progressionarithmetic Next:

(2, 4, 6, 8, …)

It is the arithmetic progression formed by all even numbers from 2. So the first term is 2, the ratio is 2, and the number of terms is 100, because we want to find the hundredth term. Look:

The_no = the₁ + (n – 1)r

The₁₀₀ = 2 + (100 – 1)2

The₁₀₀ = 2 + (99)2

The₁₀₀ = 2 + 198

The₁₀₀ = 200

By Luis Paulo Silva
Graduated in Mathematics

Source: Brazil School - https://brasilescola.uol.com.br/matematica/termo-geral-pa.htm