Sequence: Sequence and Series is one of the most important concepts in Arithmetic. A sequence refers to the collection of elements that can be repeated in any sort. In other words, we can say a sequence refers to the list of objects or items that have been arranged in a systematic and sequential manner.
Eg: a1,a2,a3, a4…….
Series: Whereas, Series refers to the sum of all the elements available or we can say A series can be referred to as the sum of all the elements available in the sequence. One of the most common examples of a sequence and series would be Arithmetic Progression.
Eg: If
a1,a2,a3, a4……. etc is considered to be a sequence, then the sum of terms in the sequence a1+a2+a3+ a4……. are considered to be a series.
Types of Sequences
Two main types of sequences are discussed here:
1. Arithmetic Sequence
Consider the following sequence for instance,
1,4,7,10,13,16…….
Here, we can see that each term is obtained by adding 3 to the preceding term. This sequence is called Arithmetic sequence or Arithmetic Progression. It is also abbreviated as A.P. Thus we can define Arithmetic Sequence as
A sequence x1,,x2,x3,……xn. Is called an Arithmetic Progression, if there exists a constant number m such that,
x2 = x1+ m
x3 = x2+ m
x4= x3+ m
.
.
xn = xn-1+ m and so on
The constant m is called the common difference of the A.P.
Thus we can write it as,
x, x+m, x+2m, x,+3m, ……x+(n-1)m |
Where x is the first term
m is a common difference.
2. Geometric Sequence
A sequence in which each term is obtained by either multiplying or dividing a certain constant number with the preceding one is said to be a geometric sequence.
For example
2,4,8,16,32,64,128…and so on
Here we can see that there is a common factor 2 between each term.
The geometric sequence can be commonly written as,
Where a is the first term
m is the common factor between the terms.
A few other types of Sequence and Series include
3. Harmonic Sequences
Harmonic Sequences refer to a series of numbers that are said to be in a harmonic sequence. These series of numbers are said to be in a harmonic sequence only if the reciprocal of all the elements that are a part of the sequence are created into an arithmetic sequence.
4. Fibonacci Number Sequence
Fibonacci Numbers are a form of a number sequence where every element can be obtained by adding two elements. With this, the sequence starts with 0 and 1. Hence, the Fibonacci Number Sequence is defined as
F1=0,
F2 = 1,
F3 = 1,
F4 = 2,
F5 = 3,
…
..
Fn = Fn-1 + Fn-2
Difference Between Sequence and Series
There is a bit of confusion between sequence and series, but you can easily differentiate between Sequence and series as follows:
A sequence is a particular format of elements in some definite order, whereas a series is the sum of the elements of the sequence. In sequence order of the elements are definite, but in series the order of elements is not fixed.
A sequence is represented as 1,2,3,4,….n, whereas the series is represented as
1+2+3+4+…..n.
In sequence, the order of elements has to be maintained, whereas in series the order of elements is not important.
Sequence and Series Formulas
Formulas for Arithmetic Sequence:
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Sequence = x, x+m, x+2m, x,+3m, ……x+(n-1)m
Where x is the first term m is a common difference.
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Common difference = m = Successive term – Preceding term = x2- x1
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General term = nth term = x+(n-1)m
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Sum of first nth terms =sn = [frac{n}{2}(2a + (n-1)d)]
Formulas for Geometric Sequence
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Sequence = a, am, am2, am3…….
Where a is the first term, m is the common factor between the terms.
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Common factor = m = Successive term / Preceding term m = [frac{am^{(n-1)}}{am^{(n-2)}}]
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General term = nth term = an = [am^{(n-1)}]
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Sum of n terms in GP = Sn = na if m = 1,
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Sum of n terms in GP = Sn = [frac{a(m^{n}-1)}{(m – 1)}] when m > 1
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Sum of n terms in GP = Sn = [frac {a(1- m^{n})}{(1 – m)}] when m < 1
Formula for Series
[s_{n}=frac{n}{2( 1 + n)}]
Sequence and Series Solved Examples
Example 1
Write an A.P when its first term is 10 and the common difference is 3.
Solution:
Step 1: Arithmetic Progression = A.P. = a, a+m , a+2m , a +3m, a+4m…….
Step 2: here, a=10 and m = 3
So let put its value in the equation
Step 3: 10, 10+3, 10 +2*3, 10 + 3*3, 10 + 4 * 3 ………
We get,
10,13,16, 19, 22, …….
Example 2
Write the first three terms of the sequence defined as an = n2 + 1
Solution:
Step1: we have an = n2 + 1
Step 2
: Putting n = 1,2,3. We get
Step 3: a1 = 12 + 1 = 1 + 1 = 2
a2 = 22 + 1 = 4 + 1 = 5
a3 = 32 + 1 = 9 + 1 = 10
Quiz Time
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Write the first five terms of each of the following sequence whose nth terms are:
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Write an A.P. having 4 as the first term and -3 as the common difference.
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Write a G.P. having 4 as the first term and 2 as the common ratio.
Summary Points
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In a math series and sequence, a is considered to be the first term, d is the common difference, and an is known to be the nth term.
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The arithmetic sequence can be explained as a, a + d, a + 2d, a + 3d, …
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Each term in a geometric progression is obtained by multiplying the common ratio of the successive term to the preceding term.
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The geometric progression formula is known as :
an = arn-1
Sn = a/(1-r) Where |r| < 1.