An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference and is denoted by d.
The nth term of an arithmetic sequence can be found using the formula an = a1 + (n-1)d, where an is the nth term, a1 is the first term, and d is the common difference.
The formula for the sum of an arithmetic series is Sn = (n/2)(a1 + an), where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
To find the sum of a finite arithmetic series, you can use the formula Sn = (n/2)(a1 + an) or you can use the shortcut formula Sn = (n/2)(2a1 + (n-1)d). Both formulas will give the same result.
Arithmetic sequences and series can be applied in various real-life situations, such as calculating interest rates, predicting population growth, and analyzing stock market trends. These concepts are also used in fields like physics, engineering, and computer science to model and solve real-world problems.