Term of arithmetic progression

Arithmetic progression is usually represented by a series in which each number compared to the previous one monotonically decreases or increases by the same step of progression. The online calculator can help find the first term of the arithmetic progression using any n term of the progression and its difference. Similarly, tasks of the format "Find the sixth term of the arithmetic progression (fifth, seventh, or any other)" .

In order to understand how the numbers of the arithmetic progression are ordered, consider the following series:
a₁
a₂=a₁+d
a₃=a₂+d=a₁+d+d=a₁+2d
a₄=a₃+d=a₁+2d+d=a₁+3d
...

It is obvious that there is a pattern in the formation of each subsequent term of the progression, which can be expressed through the previous one: a_n=a_(n-1)+d or through the first term of the arithmetic progression a₁. To find a term of the arithmetic progression through the first term, add the number of progression steps equal to n-1, where n is the ordinal number of the term of the progression that needs to be found according to the given conditions. a_n=a₁+(n-1)d

Conversely, knowing any specific n term of the arithmetic progression, you can find the first term. To do this, a special formula is derived from the previous one: a₁=a_n-(n-1)d

If the task requires finding the first terms of the arithmetic progression, then in any case, the first action should be calculating the first term of the progression, and then by adding the difference of the progression to each previous number, you can find the necessary number of first terms, for example, up to the fifth or tenth term.

The total number of terms of the arithmetic progression is by default unlimited, as the addition of the difference of the progression is an operation that can be repeated indefinitely. The limit of such a sequence will tend towards positive or negative infinity depending on the sign of the progression difference. Since the sequence will grow indefinitely, for arithmetic progression, it is possible to find the sum of the first terms or the sum of terms defined by the condition of the task.

Accordingly, knowing the sum of the arithmetic progression, finding the first term is not difficult if the formula is correctly inverted. The sum of the arithmetic progression is the arithmetic mean (from which comes the name) of the first and last terms of the progression, multiplied by the total number of terms of the progression.

The first term of the progression in this case will be equal to the doubled ratio of the sum to the total number of terms minus the last term in the sum.