An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the ‘common difference’.
To better understand, consider the sequence: 2, 4, 6, 8, 10. In this case, the common difference is 2, as each number increases by 2 from the previous one.
The general formula for the nth term of an arithmetic sequence can be expressed as: a_n = a_1 + (n – 1) * d, where:
- a_n is the nth term
- a_1 is the first term
- d is the common difference
- n is the term number
For instance, if we want to find the 5th term of the sequence mentioned earlier (2, 4, 6, 8, 10), we can use the formula:
a_5 = 2 + (5 – 1) * 2
=> a_5 = 2 + 8 = 10
In summary, arithmetic sequences form the basis for many mathematical concepts and appear in various real-life applications, from finance to engineering.