Sequences and Series

Sequences and Series: In mathematics, the sequence is a collection or list of numbers that have a logical/sequential order or pattern between them. For example, 1, 5, 9, 13, … is a sequence having a difference of 4 between each consecutive next term and each term can be represented in form 1 + 4 * ( n – 1 ) where n is the nth term of the sequence. The sequence can be classified into 3 categories:

1. Arithmetic Sequence
2. Geometric Sequence
3. Harmonic Sequence

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. Sequences can be finite or infinite, depending on whether they have a limited number of terms or continue indefinitely. The pattern or rule that defines a sequence can be arithmetic, geometric, or based on some other mathematical relationship.

The sequence in which each consecutive term has a common difference and this difference could be positive, negative and even zero is known as an arithmetic sequence.

Example:

1) 0, 2, 4, 6, …   in this sequence each and every consecutive term has a difference of 2 between them and nth term of sequence can be represented as 2 * ( n – 1 ).

2) 0, 5, 10, 15, …  is another example of arithmetic sequence with a difference of 5 between each consecutive number and nth term of sequence can be represented as 5 * ( n – 1 ).

The sequence in which each consecutive term has a common ratio is known as a Geometric sequence.

Example:

1) 1, 5, 25, 125 … in this sequence, each consecutive term have a ratio of 5 with the term before it and nth term of sequence can be represented as 5 ^{( n – 1 )}.

2) 1, -2, 4, -8, 16, … in this sequence each consecutive term have a ratio of -2 with the term before it and nth term of sequence can be represented as ( -2 )^{( n – 1 )}.

The sequence in which the reciprocal of each term forms an arithmetic sequence is known as a harmonic sequence.

Example:

(1/5), (1/10), (1/15), (1/20),… in this sequence the reciprocal of each term that is 5, 10, 15, 20, … forms an arithmetic sequence with a difference of 5 between each consecutive term.

Summation Notation is a simple method to find the sum of a sequence. Summation notation is also known as sigma notation. Sigma refers to the Greek letter sigma, Σ. The limit of the sequence is represented as shown in figure 1 where the lower limit is the starting index of the sequence and the upper limit represents the ending index of the sequence. Like as shown in figure 1 the lower limit is 1 and the upper limit is 4 so this means we need the sum of 1st,2nd,3rd and 4th term which is ( 2 * 1 ) + ( 2 * 2 ) + ( 2 * 3 ) + ( 2 * 4 ) = 2 + 4 + 6 + 8 = 20.

Summation Notation:

[Tex]\huge \sum\limits_{n=1}^4 (2*n)[/Tex]

Summation Notation for Arithmetic Sequence

Summation Notation of Arithmetic Sequence is of form  Σ (a + b * n) where a is the first term of the sequence and b is the common difference between any two consecutive terms of the sequence and therefore the nth term of the sequence would be of the form (a + (b * n)).

Example:

Let the arithmetic sequence be 0, 2, 4, 6, … so for making the summation notation we need to find the values of ‘a’ and ‘b’ where ‘a’ is the first term which is 0 so a = 0 and b is the common difference between any 2 consecutive terms which is 2 in this case, so b = 2.Therefore summation notion of sequence would be Σ (0 + (2 * n)) where the lower limit is 0 and the upper limit is ∞ as the first term of the sequence is given as 0 and ending is not defined.

Arithmetic Summation Notation:

[Tex]\huge \sum\limits_{n=0}^\infin (0\ +\ 2*n)[/Tex]

Summation Notation for Geometric Sequence

Summation Notation of Geometric Sequence is of form  Σ (a * bⁿ) where a is the first term of the sequence and b is the common ratio between any two consecutive terms of the sequence and therefore the nth term of the sequence would be of the form (a * bⁿ).

Example:

Let the geometric sequence be 2, 10, 50, 250, … so for making the summation notation we need to find the values of ‘a’ and ‘b’ where ‘a’ is the first term which is 2 so a = 2 and b is the common ratio between any 2 consecutive terms which is 5 in this case, so b = 5.Therefore summation notion of sequence would be Σ (2 * 5 ⁿ) where the lower limit is 0 and the upper limit is ∞ as the first term of the sequence is given as 0 and ending is not defined.

Geometric Summation Notation:

[Tex]\huge \sum\limits_{n=0}^\infin (2\ *\ 5^n)[/Tex]

Summation Notation for Harmonic Sequence

Summation Notation of Harmonic Sequence is of form  Σ ( 1/(a + b*n) )where a is the reciprocal of the first term of the sequence and b is the common difference between reciprocal any two consecutive terms of the sequence and therefore the nth term of the sequence would be of the form (1/(a + b * n)).

Example:

Let the arithmetic sequence be 1/2, 1/4, 1/6, … so for making the summation notation we need to find the values of ‘a’ and ‘b’ where ‘a’ is the reciprocal first term which is 2 so a = 2 and b is the common difference between the reciprocal of any 2 consecutive terms which is 2 in this case, so b = 2.Therefore summation notion of sequence would be Σ (1/( 0 + 2 * n)) where the lower limit is 1 and the upper limit is ∞ as the first term of the sequence is given as 1/2 and ending is not defined. 

Harmonic Summation Notation:

[Tex]\huge \sum\limits_{n=0}^\infin (\frac{1}{0\ +\ 2^n})[/Tex]

Example 1: Find the first 4 terms of the sequence: a_n = 2 * xⁿ + 1 and n > 0?

Solution: 

As we should take care to replace n ( and not x ) with the first 4 natural numbers as n is not equal to 0.

a₁ = 2 * x¹ + 1 

a₂ = 2 * x² + 1

a₃ = 2 * x³ + 1 

a₄ = 2 * x⁴ + 1

Example 2: Find the first 6 terms of the sequence: a_n = [Tex]\frac{1}{n\ +\ 2}[/Tex] and n ≥ 0?

Solution: 

We have to replace n by the first 6 while numbers (0, 1, 2, 3, 4, 5).

a_{0 =} [Tex]\frac{1}{0\ +\ 2} = \frac{1}{\ 2}[/Tex]

a_{1 =} [Tex]\frac{1}{1\ +\ 2} = \frac{1}{\ 3}[/Tex]

a_{2 =} [Tex]\frac{1}{2\ +\ 2} = \frac{1}{\ 4}[/Tex]

a_{3 =} [Tex]\frac{1}{3\ +\ 2} = \frac{1}{\ 5}[/Tex]

a_{4 =} [Tex]\frac{1}{4\ +\ 2} = \frac{1}{\ 6}[/Tex]

a_{5 =} [Tex]\frac{1}{5\ +\ 2} = \frac{1}{\ 7}[/Tex]

Example 3: Evaluate [Tex]\sum\limits_{n=1}^k (2*n)[/Tex]  for k = 2 and k = 4?

Solution:

For k = 2

[Tex]\sum\limits_{n=1}^2 (2*n)[/Tex] = (2 * 1) + (2 * 2) = 2 + 4 = 6 

For k = 4

[Tex]\sum\limits_{n=1}^4 (2*n) [/Tex] = (2 * 1) + (2 * 2) + (2 * 3) + (2 * 4) = 2 + 4 + 6 + 8 = 20 

Example 4: Evaluate [Tex]\sum\limits_{n=1}^4 (5^n)[/Tex]?

Solution:

[Tex]\sum\limits_{n=1}^4 (5^n)[/Tex] = (5¹) + (5²) + (5³) + (5⁴) = 5 + 25 + 125 + 625 = 780

Example 5: Evaluate [Tex]\sum\limits_{n=0}^3 (\frac{1}{n\ +\ 2^n})[/Tex]?

Solution:

[Tex]\sum\limits_{n=0}^2 (\frac{1}{n\ +\ 2^n})= (\frac{1}{0\ +\ 2^0})+(\frac{1}{1\ +\ 2^1})+(\frac{1}{2\ +\ 2^2})\\ \sum\limits_{n=0}^2 (\frac{1}{n\ +\ 2^n})=(\frac{1}{1})+(\frac{1}{3})+(\frac{1}{6})\\ \sum\limits_{n=0}^2 (\frac{1}{n\ +\ 2^n})=(\frac{6+2+1}{6})\\ \sum\limits_{n=0}^2 (\frac{1}{n\ +\ 2^n})=(\frac{9}{6})\\ \sum\limits_{n=0}^2 (\frac{1}{n\ +\ 2^n})=(\frac{3}{2})\\[/Tex]

Example 6: Write in expanded form: [Tex]\sum\limits_{n=0}^\infin (\frac{1}{n\ +\ 2^n})[/Tex] upto 4 terms?

Solution:

[Tex]\sum\limits_{n=0}^\infin (\frac{1}{n\ +\ 2^n}) = (\frac{1}{0\ +\ 2^0}) + (\frac{1}{1\ +\ 2^1})+(\frac{1}{2\ +\ 2^2})+(\frac{1}{3\ +\ 2^3})+…\\ \sum\limits_{n=0}^\infin (\frac{1}{n\ +\ 2^n}) = (\frac{1}{1}) + (\frac{1}{3})+(\frac{1}{6})+(\frac{1}{11})+…     [/Tex] 

Also Check:

Sequences and Series Class 11 Notes

Sequences and Series Class 11 NCERT Solutions

What is the difference between a sequence and a series?

A sequence is an ordered list of numbers following a specific pattern, while a series is the sum of the terms of a sequence. For example, in the sequence 2, 4, 6, 8, the series would be 2 + 4 + 6 + 8.

How do you find the nth term of an arithmetic sequence?

The nth term of an arithmetic sequence can be found using the formula: a_n = a₁ + (n – 1)d where ana_nan​ is the nth term, a1a_1a1​ is the first term, nnn is the term number, and ddd is the common difference.

What is the formula for the sum of the first n terms of a geometric series?

The sum of the first n terms of a geometric series is given by: where S_n​ is the sum, a is the first term, r is the common ratio, and n is the number of terms.

How can you determine if a series converges or diverges?

A series converges if the sum of its terms approaches a finite value as more terms are added. For a geometric series, it converges if the absolute value of the common ratio | r | < 1. For other types of series, various tests like the comparison test, ratio test, or integral test are used to determine convergence or divergence.