Sequences and Series :
In this section, we will learn about sequences and series.
Consider the following pictures.
There is some pattern or arrangement in these picture. In the first picture, the first row contains one apple, the second row contains two apples and third row contains three apples and so on.
The number of apples in each row are
1, 2, 3, ..............
These numbers are belongs to the category called sequences.
Definition of sequence :
The real valued sequence is a function defined on the set of natural numbers and taking real values.
What is term ?
Each element in the sequence is called a term of the sequence.
The element in the first position is called the first element and the element in the second position is called the second element of the sequence and so on.
What is finite and infinite sequence ?
If the number of elements in a sequence is finite then it is called a finite sequence.
If the number of elements in a sequence is infinite then it is called a infinite sequence.
A few popular sequences in math are:
What is series ?
The sum of the terms of a sequence is called series.
Let a_{1}, a_{2}, a_{3},............be a sequence of real numbers.
Then the real number
a_{1 }+ a_{2 }+ a_{3 }+ ............
is defined as a series of real number.
Special series :
There are some series whose sum can be expressed by explicit formulae. Such series are called special series.
Here we study some common special series like
(i) Sum of first "n" natural numbers.
(ii) Sum of first "n" odd natural numbers.
(iii) Sum of squares of first "n natural numbers.
(iv) Sum of cubes of "n" natural numbers.
General form of arithmetic sequence :
a, a + d, a + 2d, a + 3d,...............
General term of arithmetic sequence :
t_{n} = a + (n - 1)d
Sum of arithmetic series :
S_{n} = (n/2) [2a + (n - 1)d]
S_{n} = (n/2) [a + l]
a = First term
d = t_{2} - t_{1 }
n = number of terms
l = last term
General form of geometric sequence :
a, ar, ar^{2}, ar^{3},...........
General term of geometric sequence :
t_{n} = ar^{n-1}
Sum of geometric series :
S_{n} = a(r^{n} - 1) / (r - 1) if r > 1
S_{n} = a(1 - r^{n}) / (1 - r) if r < 1
S_{n} = na if r = 1
S_{n} = a/(1 - r) (For infinite series)
a = First term
r = t_{2} / t_{1 }
n = number of terms
Sum of first "n" natural numbers :
= n (n + 1) / 2
Sum of first "n" odd natural numbers :
= n^{2} (or) [(l + 1) / 2]^{2}
Sum of squares of "n" natural numbers :
= n(n + 1) (2n + 1) / 6
Sum of cubes of "n" natural numbers :
= [n(n + 1) / 2]^{2}
After having gone through the stuff given above, we hope that the students would have understood sequences and series.
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