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5b5. Sequences and Series

It is defined as a set of numbers specified in a particular order by some assigned law.


For example, S1 = 2, 3, 5, 7 and S2 = 1, 4, 7, 10, 13


is a
sequence of numbers where each term is three more than the previous term.

Almost in all the sequence a general term can be written which represents the terms of the sequence. For example, the general term for the sequence of consecutive odd numbers starting from 1 will be, 2n – 1, where n can take any value from 1 to infinity. Similarly the general term for the sequence of consecutive even numbers starting from 2 can be represented as 2n in principle, the general term of any sequence can be found by taking the difference between the sum of first n terms and (n – 1) terms of the sequence.


the sum of first n terms,

Series


When a sequence is represented in summation form it is called a series and the summation is represented by the symbol ‘ Σ ’.


For example, 1 + 3 + 5 + 7 + 9 is a series of first five odd numbers and is represented as,



A sequence is called an Arithmetic Progression (A.P.) if the differences between any two pairs of consecutive terms are same. This constant difference is called the common difference (d). The first term is denoted



For example, 2, 4, 6, 8, 10 or 1, 4, 7, 10, 13.


In the first one the common difference is 2 and in the second the common difference is 3.


Let the first term of an AP be ‘a’ and the common difference is ‘d’, then the terms of the AP can be written as: a, a + d, a + 2d, a + 3d...


The n term, an, is generalized as, an = a + (n – 1) d.


A sequence is called a geometric progression (G.P.) if the ratio of any term to the preceding term is a constant, called common ratio. If the first term is 'a' and the common ratio is 'r', then the sequence takes the form a, ar,










 
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