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Sequence and series is an important topic in chapter 9 of NCERT Class 11 Mathematics. Sequence is defined as a process of arranging the numbers according to specific rules applied.
- Series refers to the addition of all the numbers added in a sequence.
- It depends upon the length and number of terms.
- The number of terms added to a sequence or a series is either finite or infinite.
- It is equal to the length of the sequence.
- Arithmetic, geometric, harmonic sequences, and Fibonacci numbers are four types of sequences and series.
- The sequence is thus a particular order in which various elements follow each other.
- Series refers to the sum of all those elements.
- Sequences and series play a significant role in a variety of human activities.
- The quantity of money placed in a bank over a period of time forms a sequence.
- Populations of humans or microorganisms establish a series at different times.
Key Terms: Sequence and series, Sequence, Series, Arithmetic Sequence, Geometric sequence, Harmonic Sequence, Fibonacci Numbers, Arithmetic Progression, Common Ratio, First Term, Common Difference
What are Sequence and Series?
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Sequence and series are one of the most fundamental subjects in arithmetic. A sequence is an itemized collection of elements that allows any type of repetition, whereas a series is the sum of all elements.
- The sum of all the terms in a sequence can be used to generalize a series.
- Arithmetic progression is the common instance between sequence and series.
- The topic can further be better grasped by Important Questions for Class 11 Maths Chapter 9: Sequences and Series
Sequence
A sequence is a series of items or a set of numbers that are arranged in a specific order according to a set of rules. It is similar to sets, but the main difference is that individual terms in a sequence might appear multiple times in different positions.
- A sequence's length is equal to the number of terms.
- It is also known as progression.
- Its notion can be generalized with the indexed family.
- Rank is defined by the position of an element in the sequence.
- The rank of the first term is equal to 0 or 1.
- If the terms of a sequence are a1, a2, a3, a4,....etc., then the position of the term is 1,2,3,4,..…
- SN = a1+a2+a3 +... + aN is the corresponding series.
- Study of prime numbers and differential equation uses the concept of sequence.
Note: The series is finite or infinite depending on whether the sequence is finite or infinite.
Example of SequenceExample 1: The arrangement of seats in an auditorium or stadium by keeping a certain difference between each seat is an example of sequence. Example 2: Seating arrangement in a restaurant is another example of sequence. Example 3: Riya loves growing flowers in her garden. There is 3 flower in the 1st row, 8 flowers in the 2nd row, 13 flowers in the 3rd row, and so on. In which row did she plant 100 flowers? Ans: The sequence of the number of flowers starting from the first row is 3, 8, 13, .... Its general term is, an = n2. Substitute an = 100 here, 100 = n2 n = 10 |
Series
Series is defined as addition of terms added one after another to form a required value. In simple terms it is addition of terms in a sequence. Series are frequently stated in a compact form known as sigma notation.
- It uses the Greek symbol ∑(sigma) to indicate the summation.
- According to whether the above sequence is finite or infinite, the series is finite or infinite.
- Let us take a1, a2, a3,…,an, as a given sequence.
- The series associated with the provided sequence is therefore defined as a1 + a2 + a3 +,...+ an.
Example of SeriesExample 1: A girl studying for 1 hours every day so after 7 days, she has studied for 7 hours. Example 2: The sum of first five odd number in a series is given as: 1+3+5+7+9 = 25 |
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Sequence and Series Detailed Video Explanation:
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Types of Sequence and Series
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Sequence and series are divided into four category which are as follows:
Arithmetic Sequence and Series
An arithmetic sequence is a sequence where each term is formed by adding or subtracting a definite number from the previous one. A sequence a1 , a2 , a3 ,…, an, is called as an arithmetic sequence/arithmetic progression if
an = a + (n - 1) d
- Sum of n Terms of the series is given as:
Sn = n/2 (2a + (n - 1) d) (or) Sn = n/2 (a + an)
- where a is called the first term and d is the common difference of the A.P.
Example of Arithmetic Sequence and SeriesExample: Find the sum of the first 10 terms of the arithmetic series 1 + 4 + 7 + .…? Ans. In the given series, the first term is a = 1 and the common difference is d = 3. Using the sequences and series formulas, Sn = n/2 (2a + (n - 1) d) For the sum of 10 terms, substitute n = 10: S10 = 10/2 (2(1) + (10 - 1) 3) = 145 |
Geometric Sequence and Series
A geometric sequence is a sequence in which each term is obtained by multiplying or dividing a defined integer by the previous number. A sequence a1 , a2 , a3 , …, an , … is called a geometric progression, if each term is non-zero and ak + 1/ak = r (constant), for k ≥ 1.
- By putting a1 = a, we get a geometric progression, a, ar, ar2 , ar3 ,…. where rth value is given as:
an = a rn - 1
- Sum of the finite geometric series is given as:
Sn = a (1 - rn) / (1- r)
- Sum of infinite geometric series is given as:
Sn = a / (1 - r) when |r| < 1
- The sum is not calculated when |r| ≥ 1.
- Where a = the first term,
- r = the common ratio.
Example of Geometric Sequence and SeriesExample: Find a15 of a geometric sequence if a12 = -8 and r = ½? Ans. By the recursive formula of geometric sequence, a13 = r a12 = (1/2) (-8) = -4 Applying the same formula again, a14 = r a13 = (1/2) (-4) = -2. Therefore, a15 = -2. |
Harmonic Sequence and Series
If the reciprocal of all the elements in a sequence from an arithmetic sequence, it is said to be in a harmonic sequence. It is an infinite series where terms are arranged in a predictable pattern.
- If the terms of an arithmetic sequence and series are given as a, a + d, a + 2d, a + 3d, ....,
- In that case, the harmonic progression (or harmonic sequence) is 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d),......
- Here, a is the first term, and d is a common difference.
- Leaning tower is a common example of harmonic sequence and series.
- nth term of harmonic sequence and series is given as:
1/(a + (n - 1)d)
Example of Harmonic Sequence and SeriesExample: What is the value of the 12th term of the harmonic progression: 1/4, 1/7, 1/10, 1/13 .....? Ans. The given harmonic sequence is 1/4, 1/7, 1/10, 1/13 ........... Comparing this with 1/a, 1/(a + d), 1/(a + 2d),..., we get 1/a = 1/4, 1/(a + d) = 1/7. a = 4, a + d = 7, or d = 3. The 12th term = 1/(a + 11d) = 1/(5 + 11 x 3) = 1/(5 + 33) = 1/38. |
Fibonacci Numbers
In a fibonacci number series, each element is created by adding two preceding elements, with the sequence beginning with 0 and 1. Sequence can be stated as, F0 = 0 and F1 = 1 and sequence of nth term is given as:
Fn = Fn-1 + Fn-2
Example of Fibonacci NumbersExample: Find the 10th term of the Fibonacci series if the 8th and 9th terms are 100 and 121 respectively. Ans. We know that the 10th term is the sum of 8th term and 9th term. 10th term = 8th term + 9th term = 100 + 121 = 221 |
Arithmetic Mean
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The arithmetic mean of a list of elements is the sum of all the integers in the list divided by the number of elements in the list. Let us understand it further. We know that if any three numbers are in AP, that is, if a, b, and c are in AP.
- The difference between the first two terms a and b will be equal to the difference between the next two terms b and c.
- So now, b – a = c – b.
- Rearranging the terms,
- 2b = a + c
b = (a+c)/2
- As a result, we can claim that term b is the average of terms a and c.
- The arithmetic mean is the average in arithmetic progression.
A = S/N
- where A represents the arithmetic mean,
- N represents the number of terms,
- and S is the sum of the numbers in the list.
Multiple Arithmetic Mean between two given Numbers
Let a and b be the two numbers, and the arithmetic mean between them be A1, A2,......An.
- The resulting sequence a, A1, A2,...An, will be in AP.
- If a is the first term, then
- b = a + (n + 2 – 1 )d
- d = b−a/n+1
- So, A1 = a + (b−a/n+1),
- A2 = a + 2 *(b−a/n+1)
An = a + n *(b−a/n+1)
Example of Arithmetic MeanExample: The heights of five students are 5 ft, 6 ft, 4 ft, 8 ft, and, 2 ft respectively. Using the arithmetic mean formula, find the average (mean) height of the students? Ans. To find: Average height of the students |
Geometric Mean
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The geometric mean of two terms a and c is shown as follows: √(ac). The ratio of the two consecutive elements should be equal if a, b, and c are in geometric progression.
- b/a=c/b
- b² = ac
b = √(ac)
- Which implies b is the geometric mean of a and c
Multiple Geometric Mean between two given Numbers
Let’s say, a and b are any two given numbers and G1, G2, G3….Gn be a geometric mean between them.
Gn = a (b/a) *(m/n+1)
Example of Geometric MeanExample: Find the geometric mean of 1,2,3,4,5 Ans. The GM is given as (x1 × x2 × x3...× xn)1/n = (1 × 2 × 3 × 4 × 5)1/5 = (120)1/5 |
Sequence and Series Formulas
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The following is a list of some basic sequence and series formulas:
| Category | Arithmetic Progression | Geometric Progression |
|---|---|---|
| Sequence | a, a+d, a+2d,……,a+(n-1)d,... | a, ar, ar2,….,ar(n-1) |
| Common Difference or Ratio | Successive term - Preceding term Common difference(d)= a2 – a1 | Successive term/Preceding term Common ratio(r) = ar(n-1)/ar(n-2) |
| General Term (nth Term) | an = a + (n-1)d | an = ar(n-1) |
| nth term from the last term | an = l – (n-1)d | an = 1/[r(n-1)] |
| Sum of first n terms | sn =n2(2a + (n-1)d) | sn = a(1 – rn)1-r if r < 1 sn = a(rn - 1)r - 1 if r > 1 |
*Here, a stands for the first term, d for the common difference, r for the common ratio, n for the term position, and l for the last term.
Difference between Sequence and Series
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The difference between sequence and series is given as follows:
| Sequence | Series |
|---|---|
| Arrangement of terms in a certain pattern is known as sequence. | The sum of all the elements added in a sequence is known as series. |
| In this order of sequence is given importance. | In this order of sequence is not given importance. |
| It follows a certain pattern. | It does not follow a pattern. |
| Sequence is of the form: [pi]n/∞i=1 | Series is of the form: Sn = nr = 1 mr. |
Things to Remember
- Sequence and series is an important concepts that can be studied in NCERT Solutions for class 11 Maths Chapter 9: Sequences and Series.
- Arithmetic progression is an important concept between them.
- Sequence is defined as the definitive grouping of elements in a particular pattern.
- The formula for sequence and series can be determined for nth term and sum.
- The rate of change with respect to population is a common example of the concept.
- The patterns an individual prepare for floor tiling is another real life case of sequence and series.
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| Class 11 Mathematic Related Concept | ||
|---|---|---|
| Statistics | Geometric Probability | Types of Geometric Progression |
Sample Questions
Ques. What is the difference between finite and infinite sequences and series? (1 mark)
Ans. A finite sequence is one that includes the last term, such as a1, a2, a3, a4, a5, a6,......an. An infinite sequence, on the other hand, is never-ending, i.e. a1, a2, a3, a4, a5, a6......an.....
A finite series is composed of a finite number of terms, such as a1 + a2 + a3 + a4 + a5 + a6 +......an. The number of elements in an infinite series is not finite, i.e. a1 + a2 + a3 + a4+ a5 + a6 +....an
Ques. What are some of the most common sequence types? (1 mark)
Ans. A few popular math sequences are:
- Arithmetic Sequences
- Geometric Sequences
- Harmonic Sequences
- Fibonacci Numbers
Ques. If 4,7,10,13,16,19,22……is said to be a sequence. Find the- common difference, nth term and 21st term? (3 marks)
Ans. Here, the given sequence is, 4,7,10,13,16,19,22……
- The common difference will be = 7 – 4 = 3
- Tn denotes the nth term of the arithmetic sequence, which is given by Tn= a + (n-1)d, where “a” is the first term and d, that is the common difference.
- Tn = 4+(n – 1)3
- 4 + 3n – 3
- 3n + 1
- 64
- Here, the 21st term: T21 = 4 + (21-1)3
- 4+60 = 64.
Ques. If a given sequence is 1, 4, 16, 64, 256, 1024….. Find the: common ratio and 9th term? (2 marks)
Ans. Here, the common ratio (r) = 4/1 = 4
The preceding term is then multiplied by 4 to get the next term.
Tn denotes the nth term of the geometric sequence, which is given by
Tn = ar(n-1)
where a is the first initial term and r is said to be the common ratio.
Here, in the question, a = 1, r = 4 and n = 9
So, the 9th term is can be calculated as follows: T9 = 1* (4)(9-1)= 48 = 65536.
Ques. Find the value of the 20th term of the arithmetic sequence 5, 9, 13, 17..…? (2 marks)
Ans. The given sequence is 6, 9, 12, 15.....
The first term, a = 6
The common difference, d = 9 - 6 = 3
Using the sequence and series formulas,
an = a + (n - 1) d
For the 20th term, substitute n = 20:
a20 = a + 19d = 5 + 19*3 = 5 + 57 = 62
Ques. If the sequence 2, 4, 6…… is in AP and if each term of the sequence is multiplied by 3. Then the resultant sequence is in? (2 marks)
Ans. Given: The sequence 2, 4, 6…… is in AP, common difference d = 2 and k = 3.
Here, each term of the sequence 2, 4, 6…… is multiplied by 3.
Hence, the resultant sequence is also in AP with a common difference, K × d = 2 × 3 =6.
Ques. What will be the 13th term of the arithmetic sequence -3, -(1/2), 2…. using sequence and series formula? (2 marks)
Ans. Given a = -3, d = -(1/2) -(-3) = 5/2, n = 15
- Using the formula for nth term of an arithmetic sequence:
- an = a+(n-1)d
- Putting the known values:
- a13 = -3 +(13-1) 5/2
- a13 = 27
Ques. Find the value of the 21st and the 22nd terms in the Fibonacci series given that the 19th and 20th terms in the series are 25 and 40? (2 marks)
Ans. Using the Fibonacci series formula, we can say that the 21st term is the sum of 19th term and 20th term.
21st term = 19th term + 20th term = 25 + 40 = 65
Now, 22nd term = 20nd term + 21st term = 65 + 40 = 105
Ques. Find the sum of the first 20 terms of the arithmetic series 1 + 3 + 6 + .…? (2 marks)
Ans. In the given series, the first term is a = 1 and the common difference is d = 3.
Using the sequences and series formulas,
Sn = n/2 (2a + (n - 1) d)
For the sum of 20 terms, substitute n = 10:
S10 = 20/2 (2(1) + (20 - 1) 3) = 590
Ques. Find the 12th term of the harmonic progression, if the fifth term is 1/10, and the eighth term is 1/28? (2 marks)
Ans. The fifth term = 1/(a + 4d) = 1/10
The eight term = 1/(a + 7d) = 1/28
Comparing this we have a + 4d = 10, and a + 7d = 28.
Solving these above two expressions we have a = -14, and d = 6.
The 12th term of the harmonic progression = 1/(a + 11d) = 1/(-14 + 6x3) = 1/(4)
Ques. Find the sum of the first 10 terms of the geometric sequence 4, 8, 16, ..…? (2 marks)
Ans. Here, the first term is, a = 2.
The common ratio, r = 4/2 = 8/4 = 16/8 = ... = 2.
Number of terms is, n = 10.
The sum of finite geometric sequence formula is,
Sn = a(rn - 1) / (r - 1)
S10 = 2 (210 - 1) / (2 - 1)
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