Arithmetic Progression First Term Calculator

This Arithmetic Progression Calculator is one of two specialist calculators designed to calculate arithmetic progression based on specific known criterea. The Arithmetic Progression First Term Calculator will calculate:

1. The sum of the first n-terms of an arithmetic series when the first term and the common difference are given.

You can also calculate the sum of the first n-terms of an arithmetic series when any two terms of the series are given using this Arithmetic Progression Calculator

Arithmetic Progression First Term Calculator Parameters: The number of terms is a natural (counting) number.

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each arithmetic progression first term calculation, the Arithmetic Progression First Term Calculator will automatically calculate the results and update the formula elements with each element of the arithmetic progression first term calculation. You can then email or print this arithmetic progression first term calculation as required for later use.

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The Arithmetic Progression First Term Calculator has practical application and use in the following fields and disciplines

Engineering
Math
Physics

Theoretical description

An arithmetic progression is a special type of number sequence where the difference between any two consecutive terms is always constant. This common difference is denoted by d, while the terms of the arithmetic progression are denoted by x₁, x₂, x₃, , x_n (or a₁, a₂, a₃, , a_n), where x₁ (or a₁) is the first term, x₂ (or a₂) the second term, and so on. The n-term x_n (or a_n) is known as the general term of the arithmetic progression.

For example, 4, 7, 10, 13, is an arithmetic progression where the first term is x₁ = 4, the second term is x₂ = 7, and so on. The common difference d is obtained by subtracting two consecutive terms, for example, x₂ - x₁, where in this specific case, we have d = 7 - 4 = 3.

In general, we have

The last row gives the general formula for the n-th term of an arithmetic progression. It is

Another thing we can do with an arithmetic progression is to find the sum of the first n-terms. For this, we use a formula known as the Gauss Law. We can derive this formula by taking the sum of the first n-terms of an arithmetic progression in two ways: (1) in terms of x₁, i.e. from the first to the last, and (2) in terms of x_n, i.e. from the last to the first. We have

and

Writing everything in terms of x₁ in the first formula and in terms of x_n in the second formula yields

and

Adding the two formulas brings the cancelling out of all d-related terms. In this way, we obtain

We can write this formula in a shorter way as

or

The last formula is the mathematical representation of the Gauss Law. It allows us to calculate the sum of the first n-terms of an arithmetic progression when the first term and the common difference are known.

For example, if we want to calculate the sum of the first 50 terms (n = 50) of the arithmetic progression 6, 13, 20, (x₁ = 6 and d = x₂ - x₁ = 13 - 6 = 7), first we find the 50th term (x₅₀) using the formula

Then, we calculate the sum of the first 50 terms of this progression through the Gauss formula

Thus, we have

and

It is clear that any attempt to calculate the sum of these terms one by one would require a lot of time and efforts.

If the first term is unknown but we know two other terms x_p and x_q of the sequence, where p and p show the number of the corresponding term, first we calculate the common difference using the formula

then, we use the approach explained earlier to find whatever else is required. For example, if in an arithmetic progression are given only x₈ = 23 and x₁₄ = 35 (p = 8 and q = 14) and the sum of the first 30 terms is required (n = 30), first we calculate the common difference d through the formula

Next, we calculate the first term given that x₈ = 23 and d = 2 are already known. Thus, for n = 8, we obtain

The next step involves finding the 30th term (n = 30). We use the same formula as above to find it. Thus,

Last, we calculate the sum of the first 30 terms (n = 30) for which the question is interested. We use the Gauss Law formula for this purpose. Thus,

All the above reasoning can be extended to include other numbers as well, not just integers. Only the number of terms n must be a natural number.