Introduction to Sequence

A set of numbers where the numbers are arranged in a definite order, like the natural numbers, is called a sequence.

We write numbers 1, 2, 3, 4, . . . in an order. In this order we can tell the position of any number. For example,  number 13 is at 13th position. The numbers 1, 4, 9, 16, 25, 36, 49, . . . are also written in a particular order. Here 16 = 4² is at 4^th position. similarly, 25 = 5² is at the 5^th position; 49 = 7² is at the 7^th position. In this set of numbers also, place of each number is detremined.

In a sequence a particular number is written at a particular position. If the numbers are written as a₁,a₂,a₃,a₄,..... then a₁ is first, a₂ is second, . . . and so on. It is clear that an is at the nth place. A sequence of the numbers is also represented by alphabets f₁, f₂, f₃, . . . and we find that there is a definite order in which numbers are arranged.When students stand in a row for drill on the playground they form a sequence.We have experienced that some sequences have a particular pattern.Complete the given pattern

Look at the patterns of the numbers. Try to find a rule to obtain the next number from its preceding number. This helps us to write all the next numbers. See the numbers 2, 11, -6, 0, 5, -37, 8, 2, 61 written in this order.
Here a₁ = 2, a₂ = 11, a₃ = -6, . . . This list of numbers is also a sequence. But in this case we cannot tell why a particular term is at a particular position ; similarly we cannot tell a definite relation between the consecutive terms.
In general, only those sequences are studied where there is a rule which determines the next term.

For example (1) 4, 8, 12, 16 . . . (2) 2, 4, 8, 16, 32, . . . (3)`1/5,1/10,1/15,1/20,........`