Sets, Sequences, and Series

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1.
a
Positive integers (n) = 1, 2, 3, 4, 5, 6, . . .
Let,
A = Set of positive integers
A is then represented as:
A = {1, 2, 3, 4, 5, . . . }
2.
c
Set: A collection ofunique objects (elements),so elements cannot be re-
peated.
eg.A = {1, 2, 3} is a set but A = {1, 2, 2, 3} is not a set.
Sequence:An ordered list of elements which can also be infinite.
In a sequence the elements can be repeated unlike in sets.
eg. {0, 1, 1, 2, 3, 5, 8, . . . } is a sequence (Fibonacci) but it cannot be termed
as set.
d
Linear sequence:A sequence which increases or decreases by the same amount
each time is called a linear sequence.Alternatively, if the difference between
the consecutive terms in a sequence remains same it is a linear sequence.
Let, A = (a0, a1, a2, a3, a4, a5, . . . , an, an+1, . . . ) be any sequence.If the dif-
ference between consecutive terms:|a1 a0| = |a2 a1| = · · · = |an+1 an|,
is constant and equalto the same amount d,where d is any realnumber,
then A is a linear sequence.
Eg. (1, 2, 3, 4), (1, 3, 5, 7, . . . ) etc.
e
5, 8, 11, 14, . . .
Difference between the consecutive termsof the sequence is:|8 5| =
|11 8|= |14 11| . . . ,which is 3. Therefore,the common difference is
1
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3.
f
5, 8, 11, 14, . . .
5 is the first term n = 1, 8 is the second term n = 2 and so on.
Let the nth term in the sequence be represented as Un = an+b.To determine
the nth rule, we need to determine a and b.
For n = 1:U1 = 5 = a · 1 + b, implies a + b = 5
For n = 2:U2 = 8 = a · 2 + b, implies 2a + b = 8
Therefore, we have:
a + b = 5 (1)
2a + b = 8 (2)
Solving (1) and (2) simultaneously, we get:a = 3 and b = 2
Implies,
Un = 3n + 2
is the nth rule for the sequence.
g
The 20th term (n = 20) in the sequence is determined using the nth rule:
Un = 3n + 2
Therefore, the 20th term is:
U20 = 3(20) + 2 = 62
3
h
Series:Results from adding all the terms of the sequence.
Given sequence is:
5, 8, 11, 14, . . .
2
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Its corresponding series is:
5 + 8 + 11 + 14 + . . .
i
In an arithmetic series, the difference between consecutive terms of the series
remains constant and equal.As it was already determined that the common
difference is 3 for the given sequence, it implies that the series obtained in h
is an arithmetic series.
j
The nth term of the series/sequence is:
Un = 3n + 2
Summation of the first 20 terms of the sequence is given by:
S =
20X
i=1
Un =
20X
i=1
3n + 2
k
The given series is:
3 + 9 + 27
Common ratio (r) for a series is the ratio of consecutive terms:
r = 9
3 = 27
9 = 3
Therefore, the common ratio is 3.
l
The nth of a geometric series is determined using formula:
Un = a0(r)n−1
where, a0 is the leading term of the series.Since the series starts from 3, we
have a0 = 3 and as determined earlier r = 3.
Therefore, the nth term in the series is given by:
Un = 3(3)n−1
3
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m
Formula for sum of first n terms of geometric series:
Sn = a0
rn 1
r − 1
For the given series:a0 = 3 and r = 3.Therefore,the sum of first 6 terms
(n = 6) of the series is:
S6 = 3 ×36 1
3 1= 3 ×728
2 = 1092
References:
1. Scocco,D. (2012)Sets,Sequencesand Tuples.’Programming Logic.
www.programminglogic.com
https://www.programminglogic.com/sets-sequences-and-tuples/
2. Stapel, E. (2019) ‘Geometric Series.’ Series.Purple Math.
https://www.purplemath.com/modules/series5.htm
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