Imagine that you are in a pizza outlet and you have ordered a pizza. After taking your order, the waiter has provided you with an order number 280. Currently, they are preparing pizza for order number 270. So, how many orders will be served before your one? As you are in a sequence, 9 orders will be served before you. To comprehend this well, let us learn sequences and series right now.
Sequence refers to a specially ordered number list. It is a chain of numbers that follows a specific pattern, and all these numbers of a sequence are known as terms. For instance, a sequence looks like this:
[ 1, 3, 5, 7, 9, 11,....]
This is a very straightforward sequence, and everyone can understand that it is a collection of odd numbers. Here the number of terms does not have a limit or is infinite. Hence, a sequence containing an infinite number of terms is called an infinite sequence. On the other hand, when number of terms in a sequence becomes countable, it is known as a finite sequence. Here is an example:
[1, 3, 5, 7, 9, 11, ......., 133]
As you can determine or count the number of terms, it is a finite sequence.
A sequence is usually referred by A or S and terms of the same are named by ai or an. Here “i” or “a” is the counter or index. So accordingly, the second term of a sequence can be written as a2 and a12 would mean the twelfth term.
Sequences can also be expressed on the basis of its terms like:
(ai)ni = 1
For instance, if the beginning value is 3 and continuing till infinity, it can be stated as:
(an)n ∞ =3
When you sum up all the terms of a sequence, it is called a series. A series is said to be finite or infinite depending on whether the sequence is finite or infinite. It is expressed as sigma, which denotes the involvement of summation. Following is an example of a series:
If [1, 3, 5, 7, 9, 11] is a sequence then [1 + 3 + 5 + 7 + 9 + 11] is the corresponding series. It can also be represented as:
S = Sum (1, 3, 5, 7, 9, 11)
Next, let us proceed with the types of sequence and series.
Sequence and Series Types
Some common types of sequences and series are:
Let us understand them individually.
In this sequence, the difference between any two back to back terms is constant. For example 2, 6, 10, 14,.... is an arithmetic progression as 4 is the difference between any two consecutive terms.
This difference at every step is called common difference and is denoted by “d”.
In a geometric progression or sequence, every term after the first one is obtained by multiplying or dividing by a fixed non-zero value. For example 1, 3, 9, 27,.... is a geometric sequence because at each stage the terms are multiplied by 3. On the other hand 9, 3, 1, 1/3,.... is also geometric because over here each stage is divided by 3. The number which is multiplied or divided at every step is known as common ratio and written as “r”.
In this type of sequence, the reciprocal of all the terms are in an arithmetic sequence. For example 1, ½, 1/3, 1/4,..... is harmonic progression because 1, 2, 3, 4 are just counting numbers and difference between two numbers (here 1) is same.
If you want to learn more on the chapter sequence and series class 11, you can refer to our comprehensive study materials. Moreover, you can download the Vedantu app to know various sequence and series aptitude questions and improvise your mathematical skills.
1. What are the sequences and series equations to find nth term for arithmetic, geometric and harmonic progressions? Also, write the formula for harmonic mean.
Ans. The sequence and series formulas with respect to arithmetic progression for nth term is an = a+(n-1)d
For geometric progression, the formula is an = ar(n-1).
For harmonic sequence, the formula is Tn = 1/ (a + (n-1)d).
Lastly, the formula for harmonic mean is 1/H = 1/n (1/a1 + 1/a2 +...+1/an)
2. What is Fibonacci series?
Ans. In the case of Fibonacci series, the first two terms remain fixed, but from the third term, every term will be the summation of preceding two terms. Here, every term is represented as an where n is the series’ nth term. See the example below to understand it properly.
If the first term is 1, then the second term is also 1, and the third term will be a1 + a2, which is 1 + 1 = 2.
Same way the fourth term a2 + a3 = 1 + 2 = 3 and so on.
So, the Fibonacci series can be written as [1, 1, 2, 3, 5, ....]
3. What is the next term and common difference of the sequence: 3, 11, 19, 27, 35, ....?
Ans. In order to determine the common difference, subtraction of consecutive two terms is necessary. So 11 – 3 = 8, 19 – 11 = 8, 27 – 19 = 8 and 35 – 27 = 8
The difference 8 remains constant for every pair, therefore common difference d = 8.
Since the sequence holds five terms, the sixth term is required to be determined.
To find the next term, you will have to add the fifth term with d.
35 + 8 = 43
So, common difference is 8, and next term is 43.
4. What will be the common ratio and 9th term of the sequence 1, 4, 16, 64, 256, 1024?
Ans. The common ratio is 4/1 = 4.
In case of a geometric progression, the previous term must be multiplied by 4 to get the following term.
The nth term can be signified by Tn, and the expression is Tn = ar(n-1), where a and r are the first term and common ratio respectively.
Therefore the 9th term can be evaluated as T9 = 1 x (4) (9-1) = 48 = 65536