Difference Between Series and Sequence

Series vs Sequence

Though the words series and sequence are common words of English language they find interesting application in mathematics where we encounter series and sequences. Students do not understand the difference between series and sequence and sometimes pay dearly with their marks being deducted when they use these terms incorrectly. This article will differentiate between a series and a sequence to remove all doubts in the minds of readers.

Mathematicians all over the world have been fascinated with the behavior of sequences and series. It is amazing to see the works of great mathematicians like Cauchy and Weierstrauss as these genius men studied complex sequences and series with just paper and pen what many modern mathematicians cannot even think of attempting with computers and calculators.

Let us see what a sequence is. Well, as the name implies, a sequence is an orderly arrangement of numbers. There are sequences with random numbers, but mostly sequences have a definite pattern that is used to arrive at the terms of the sequence. Sequences can be pure arithmetic or geometric sequences.

Arithmetic sequence

If a sequence of values follows a pattern of adding a fixed amount from one term to another, it is called an arithmetic sequence. The number that is added to get to the next term of the sequence remains constant. This fixed amount is called the common differences, referred to as d , and it can be easily found by subtracting first term from second term of the sequence. Here are some examples of arithmetic sequences

1, 3, 5, 7, 9, 11 …

20, 15, 10, 5, 0, -5 …

The formula to find any term of the sequence is

an = a1 + (n-1)d

And the formula to find the sum of any terms of the sequence is

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

A special type of sequence is a geometric sequence where terms are found by multiplying with a common difference.

2, 4, 8, 16, 32…

Here, next term is obtained not by adding but multiplying by 2. There are many more types of sequences that are a subject of study by mathematicians.

A series is the summation of a sequence. So if you have a finite sequence made up of numbers, you get series when you add up individual terms. Series can be found for infinite sequences also.

Series vs Sequence

• Sequence and series are encountered in mathematics

• Sequence is an arrangement of numbers in an orderly manner.

• Sequences are of many types and most popular are arithmetic and geometric

• Series is the sum of a sequence which one gets when he adds up all individual numbers of a sequence.