Arithmetic sequence

An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed amount (called the common difference) to the preceding term. For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3.

Arithmetic sequences have many important applications in mathematics and beyond. They are used in algebra, geometry, physics, and many other fields.

One of the most important properties of arithmetic sequences is that they can be easily described using a formula. Specifically, the nth term of an arithmetic sequence can be found using the formula:

an = a1 + (n-1)d

where an is the nth term of the sequence, a1 is the first term of the sequence, n is the number of the term you want to find, and d is the common difference.

For example, let's find the 10th term of the sequence 2, 5, 8, 11, 14, ... using the formula above. Here, a1 = 2, n = 10, and d = 3. Plugging these values into the formula, we get:

a10 = 2 + (10-1)3
a10 = 2 + 27
a10 = 29

So the 10th term of the sequence is 29.

Another important property of arithmetic sequences is that the sum of the first n terms can also be found using a formula. Specifically, the sum of the first n terms of an arithmetic sequence can be found using the formula:

Sn = (n/2)(a1 + an)

where Sn is the sum of the first n terms of the sequence.

For example, let's find the sum of the first 5 terms of the sequence 2, 5, 8, 11, 14, ... using the formula above. Here, a1 = 2, n = 5, and d = 3 (which we can use to find a5). Plugging these values into the formula, we get:

a5 = 2 + (5-1)3
a5 = 2 + 12
a5 = 14

Now we can use the formula for the sum of the first 5 terms:

S5 = (5/2)(2 + 14)
S5 = (5/2)(16)
S5 = 40

So the sum of the first 5 terms of the sequence is 40.

Arithmetic sequences are used in many real-world applications. For example, they can be used to model linear relationships, such as the growth of a population or the depreciation of an asset over time. They are also used in finance to calculate the future value of an investment or the amount of a loan payment.

In summary, arithmetic sequences are an important mathematical concept with many applications in various fields. They are easily described using formulas and can be used to model real-world phenomena.