What is an Arithmetic Sequence?

Arithmetic is simple mathematics, concerned with the main basic operations: Addition, subtraction multiplication and division. The term refers to basic number theory. Other branches of mathematics include geometry, and algebra. The sequences of numbers in arithmetic formulae are important and follow a few basic rules. 

An Arithmetic Sequence, however, refers specifically to numbers, placed in order, with a constant difference between them. There are no complex calculations in an arithmetic sequence. This is also called an Arithmetic Progression (AP). 

In general, a sequence is a set of things placed in order. An arithmetic sequence is a specific kind of series, where the difference between the numbers is constant. This constant difference may be negative, or positive. 

To calculate an arithmetic sequence then, we require the first term, which we call a, and the difference (d); which is constant between terms in the case of an arithmetic sequence. The difference is the second term (T2) minus the first term (T1).

Understanding this, we can calculate the general term for an arithmetic sequence.

T1 = a
T2 = T1 + d = a + d
T3 = T2 + d = (a + d) + d = a + 2d
T4 = T3 + d = (a + 2d) + d = a + 3d
…
Tn = Tn-1 + d = (a + (n-2-)d) + d = a + (n -1) d

And so the general term for the nth term is:
Tn = a + (n-1) d

Arithmetic sequences can continue to infinity, or n, the number of terms may be a fixed number. 

Real life applications of Arithmetic Sequence:

This is a number pattern which you may frequently incur in real life, but there are some applications, nonetheless. Imagine you’re putting a set amount into a piggy bank each day. You started with $15 in the piggy bank and you put $3,50 in each day. In this case, a = $15 and d = $3,50.

So you know that after 80 days, you will have
T80 = 15 + (80 - 1)*$3,50

Using our calculator, you can see that you will have $291,50.
Remember the key is that the difference is constant. If you rather put this into an interest-earning savings account, or if the amount you deposit daily varies, then it’s not an arithmetic sequence.

For more complex problems where d is not a constant, but grows, or declines at a constant rate, we would rather consider geometric sequence.

More on the term d – the constant difference:
Let’s look a bit closer at d, the difference term in an arithmetic sequence:

	d can be positive or negative
	d can be a natural number, or a fraction
	d can be zero, and in this case we have a monotone sequence, where since you’re adding or taking away nothing, all the terms are equal to the previous term.

Arithmetic Series

An Arithmetic Series is the summation of all the terms in the sequence, also knows as the sum of a finite Arithmetic Progression. This is where the calculator comes in particularly handy, because manually calculating this for many terms using only a, and d could take some time!
If you have few terms, as in the following example, this is manageable:

1+3+5+7 = 16

In this example, a = 1 and d = 2.

But for more large sequences, it is useful to understand the general term for the Arithmetic Series.

Imagine a sequence,  where the first term = a, and the final term in the sequence = l.

 We know from the Arithmetic Sequence that the terms of the sequence can be shown as follows:
T1 = a
T2 = a + d
T3 = a + 2d
…
Tn = a + (n -1)d

To calculate the Arithmetic Series, we take the sum if all the terms of a finite sequence:

∑_(n=1)^l▒〖Tn=Sn〗

The Sum of all terms from a1 (the first term) to l the last term in the sequence, where l = an

Now remember that sequences have a constant d, or difference. Therefore, by summing the first and last term first, we find that what we are adding, we’re taking away, it is like a constant moving point through the data in the sequence. 

Sn = a + (a + d) + (a +2d) + … + (l-2d) + (l-d) + l
The way we illustrate the rule or pattern is by adding the same sequence in reverse. 

Sn = a + (a + d) + (a +2d) + … + (l - 2d) + (l - d) + l
+ Sn = l + (l-d) + (l – 2d) + … + (a + 2d) + (a + d) + a
2Sn = (a + l) + (a + l) + (a + l) + …+ (a + l) + (a + l) + (a + l) 

2Sn = n x (a + l)
 Sn = n/2 x (a + l)

Now, we know, that l, the last term, is 
l = Tn = a + (n-1)d, so we substitute this:

Sn = n/2 x (a + [a + (n -1)d])
Sn = n/2 x [2a + (n -1)d]
This is then the general formula for sum of an Arithmetic Sequence, the Arithmetic Series.

Another way to understand the difference:
Arithmetic Sequence: {2, 4, 6, 8, 10}
Arithmetic Series: S = {2 + 4 + 6 + 8 + 10}

For example, let’s return to the example of the piggy bank. You want to know how many days you need to put your set amount into the piggy bank before you have $500. We’re solving for n:

500 = n/2 (2(15) + (n – 1)3,50)
500 = n/2 (30 + (n – 1)3,50)
1000 = n(30 +3,50n – 3,50)

Using the quadratic formula you’d get that 
n = -21,11  or  n = 13,53
Since we know in reality, it can’t be a negative number, since we’re searching for several payments, the answer is 13,53. You’d have saved the amount in 2 weeks!
So if you know the first term, and the number of terms, you can find the value of the nth or last term, and also find the number of terms (n). Or just use our handy calculator.

What about an Arithmetic series to infinity?

It seems intuitive that the sum of an infinite number of terms is infinity in any direction, except where d = 0. For Geometric Series, the sum may be a finite number.
A Geometric Progression differs from an Arithmetic Progression because the difference if not constant. One example of this is exponential growth and decay. BigData and the Information Age.