What does sequence mean?

Arithmetic is a branch of mathematics that deals usually with the nonnegative real numbers including sometimes the transfinite cardinals and with the appliance of the operations of addition, subtraction, multiplication, and division to them. The basic operations under arithmetic are addition, subtraction, division, and multiplication. Progression may be a list of numbers (or items) that exhibit a specific pattern.

What does Sequence mean?

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters or in other words a particular order in which related things follow each other. A sequence is an appointment of any objects or a group of numbers during a particular order followed by some rule. There are many types of sequences but mostly four types of sequences are well known, lets take a look at these 4 types,

• Arithmetic Sequences
• Geometric Sequences
• Harmonic Sequences
• Fibonacci Numbers 

Arithmetic Sequences  

In an arithmetic sequence, the difference between consecutive terms is always the same. For example, the sequence 5, 9, 13, 17… is arithmetic because the difference between consecutive terms is always four. The difference between a sequence and a progression is that to calculate its  nth term, a progression has a specific formula i.e,

 T_n = a + (n-1)d

Which is the formula of the n^th term of an arithmetic progression.

Geometric Sequences 

A geometric sequence goes from one term to the next by always multiplying or dividing by the same value. The number multiplied (or divided) at each stage of a geometrical sequence is named the common ratio. The formula for the nth term of the geometric sequence is, where a₁ is the first term, r is the common ratio, and an is the n^th term,

a_n= a₁ r^n-1

Harmonic Sequences 

Harmonic sequence, in mathematics, a sequence of numbers a₁, a₂, a₃,… such their reciprocals 1/ a₁, 1/ a₂, 1/ a₃,… form an arithmetic sequence (numbers separated by a common difference). The arithmetic sequence is just the reciprocal of the harmonic sequence. The nth term for the harmonic sequence where T_n is the nth term, n is the number of terms, and d is a common difference,

Fibonacci Numbers 

The Fibonacci series 1, 1, 2, 3, 5, 8, … is an example of a sequence. The Fibonacci sequence is basically a sequence where the next term is the sum of the earlier 2 terms starting with 1.

Sample Problems

Question 1: Find the 17th term of the following arithmetic progression 6, 10, 14, 18, 22, 26, …

Solution:

Formula of nth term of an A.P. is  Tn = a + (n-1)d

Here, a = 6 and d=(10-6)=4

Therefore , 17th term = 6 + (17-1)×4

=6 + 16 × 4 = 6 + 64 = 70

Question 2: Find the sum of the following arithmetic progression  2, 7, 12, 17, 22, …. , 52

Solution:

Formula of sum of an A.P. when first and last term is given is : [n/2](a+l)

Here , a = 2 , d = 5 and l = 52

Tn = a + (n-1)d

Therefore , 52 = 2 + (n – 1 ) × 5

 52 = 2 + 5×n -5 

52 + 3 = 5× n

55/5 = n

 n = 11

Therefore , sum = (11/2) ×(2+52)

=11/ 2 ×54

=11 × 27

=297

Question 3: Find the 10th term of a Fibonacci series.

Answer:

 In a Fibonacci series, the next term is sum of previous two therefore ,

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233

Therefore, 10th term is 55 .

Question 4: Is the given series a Geometric progression: 2, 4, 8, 32, 64, 128 

Solution:

In a geometric progression, the common ratio is a fixed number but in this series we have two common ratio as 4 /2 = 2 and 32/8 = 4

Therefore, it’s not a geometrical progression .

Question 5: Find the common ratio of the following series: 3, 6, 12, 24, 48, …

Solution:

Common Ratio = current term / its preceding term

= 12 / 6

= 2