SEQUENCE & SERIES

Sequence & Series

A sequence is a pattern of numbers arranged in a particular order. Each of the number in the sequence is called a term. The terms are related to one another according to a well defined rule.

Consider the sequence 1, 4, 7, 10, 13 …., 1 is the first term,(T1) 4 is the second term(T2), 7 is the third term (T3).

The sum of the terms in a sequence is regarded as series. The series of the above sequence is

1 + 4 + 7 + 10 + 13 = 35

 

The nth term of a Sequence

The nth term of a sequence whose rule is stated may be represented by Tnso that T1, T2, T3etc represent the first term, second term, third term … etc respectively.

Consider the sequence 5, 9, 13, 17, 21 ……..

T1 = 5 + 4(0)

T2 = 5 + 4(1)

T3 = 5 + 4 (2)

T= 5 + 4 (3)

Tn = 5 + 4 (n – 1)

Tn = 5 + 4n – 4       = 4n +1

when n = 30

T30 = 4(30) + 1

T30 = 121

Find the nth term of these sequences:

(i)                  3, 5, 7, 9 …… 2n + 1

(ii) 0, 1, 4, 9 ……… (n -1)2    

(iii) 1/3, 3/4, 1, 7/6 ………………2n – 1

                                                                n + 2

Arithmetic Mean

If a, b, c are three consecutive terms of an A.P, then the common difference, d, equals

b – a = c – b    = common difference.

b + b = a +c

2b = a + c

b = ½(a +c)

Examples

(i)         Insert four arithmetic means between -5 and 10.

(ii)         The 8th term of a linear sequence is 18 and the 12th term is 26. Find the first term, the common difference and the 20th term.

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