# Arithmetic Progression First Term Calculator

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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.

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

 🖹 Normal View🗖 Full Page View Calculator Precision (Decimal Places)0123456789101112131415 First term of the arithmetic progression (x1) Common difference (d) Total number of terms (n)
Arithmetic Progression Calculations and Formula Sum of the first n-terms (Sn) = Sn = [2 ∙ x1 + (n - 1) ∙ d] ∙ n/2 Sn = [2 ∙ + ( - 1) ∙ ] ∙ /2 Sn = [2 ∙ + ∙ ] ∙ /2 Sn = ∙ /2 Sn = /2 Sn = First term of the arithmetic progression (x1) = Common difference (d) = Total number of terms (n) =

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

## 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, , xn (or a1, a2, a3, , an), where x1 (or a1) is the first term, x2 (or a2) the second term, and so on. The n-term xn (or an) 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

x2 = x1 + d
x3 = x2 + d = x1 + 2d
x4 = x3 + d = x1 + 3d

xn = xn - 1 + d = x1 + (n - 1) ∙ d

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

xn = x1 + (n - 1) ∙ d

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 xn, i.e. from the last to the first. We have

Sn = x1 + x2 + x3 + ⋯ + xn

and

Sn = xn + xn - 1 + xn - 2 + ⋯ + x1

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

Sn = x1 + (x1 + d) + (x1 + 2d) + ⋯ + [x1 + (n - 1)d]

and

Sn = xn + (xn - d) + (xn - 2d) + ⋯ + [xn - (n - 1)d]

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

Sn + Sn = (x1 + xn ) + (x1 + xn ) + ⋯(x1 + xn )n times

We can write this formula in a shorter way as

2Sn = (x1 + xn ) ∙ n

or

Sn = (x1 + xn ) ∙ n/2

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

xn = x1 + (n - 1) ∙ d

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

Sn = (x1 + xn ) ∙ n/2

Thus, we have

x50 = 6 + (50 - 1) ∙ 7
= 6 + 49 ∙ 7
= 6 + 343
= 349

and

S50 = (6 + 349) ∙ 50/2
= 8875

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 xp and xq of the sequence, where p and p show the number of the corresponding term, first we calculate the common difference using the formula

d = xq - xp/q - p

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

d = xq - xp/q - p
= x14 - x8/14 - 8
= 35 - 23/14 - 8
= 12/6
= 2

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

xn = x1 + (n - 1) ∙ d
x1 = xn - (n - 1) ∙ d
= 23 - (8 - 1) ∙ 2
= 23 - 7 ∙ 2
= 23 - 14
= 9

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

xn = x1 + (n - 1) ∙ d
x30 = x1 + (30 - 1) ∙ d
x30 = 9 + (30 - 1) ∙ 2
= 9 + 29 ∙ 2
= 9 + 58
= 67

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,

Sn = (x1 + xn) ∙ n/2
Sn = (9 + 67) ∙ 30/2
= 76 ∙ 30/2
= 1140

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.

## Sequences and Series Math Tutorials associated with the Arithmetic Progression First Term Calculator

The following Math tutorials are provided within the Sequences and Series section of our Free Math Tutorials. Each Sequences and Series tutorial includes detailed Sequences and Series formula and example of how to calculate and resolve specific Sequences and Series questions and problems. At the end of each Sequences and Series tutorial you will find Sequences and Series revision questions with a hidden answer that reveal when clicked. This allows you to learn about Sequences and Series and test your knowledge of Math by answering the revision questions on Sequences and Series.

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