What is a sequence? What is an AP?
Answers
sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members. The number of elements is called the length of the sequence.
AP
an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2.
Hi,
Here is your answer,
Question = What is a Sequence ?
Answer = Sequence is a function whose domain is a subset of Natural Numbers. It represents the images of 1,2,3,.........,n as f₁ , f₂ , f₃,..........,fn where fn = f(n)
⊕ NOTE = Now, let me give you a example related to Sequence so that you will have a better understanding and a clear idea of this topic.
Example :- So, let us assume there is a generation gap of 30 years, we are asked to find the numbers of ancestors, i.e parents, grandparents and great grandparents etc. that a person might have over 300 years.
⊕ Here, the total number of Generations = 300/30 = 100
↔ The number of person's ancestors for the 1st, 2nd , 3rd ,..........,10th generations. Therefore, these numbers form 2 , 4, 8, 16 , 32,........, 1024.
↔ These numbers form what we can as ''SEQUENCE''. Now, let us consider the successive quotients that we obtain in the division of 10 by 3 at different steps of division. In his process we get 3, 3.3 , 3.33 , 3.333 , ........... and so on.
↔ These quotients also form a sequence. The various numbers occurring in a sequence are called it's ''TERMS''.
↔ Here, we donate the terms of a sequence by a, a₁ , a₂ , a₃,.........,an etc, the subscripts denote the positions of the term. The nth term is the number at the nth position of the sequence and is denoted by an. The nth term is also called the ''GENERAL TERM'' of the sequence. Thus, the terms of the sequence of person's ancestors mentioned above are:-
a₁ = 2 , a₂ = 4 , a₃ = 8,.........., a₁₀ = 1024
Now, similarly in the example of successive quotients
a₁ = 3 , a₂ = 3.3 , a₃ = 3.33 ,..........., a₆ = 3.33333, etc.
I hope you have understood what is sequence.
Question = What is an A.P ?
Answer = A sequence a, a₁ , a₂ , a₃,.........,an is called arithmetic progressions /sequence if an+1 + d, n ∈ N, where a₁ is called the first term and the constant term d is called the common difference of an A.P
→ Now, let us consider an A.P ( in it's standard form) with first term (a) and common difference (d)
a , a + d , a + 2d,..........
⊕ Then the nth (general term) of the A.P is an = a + (n-1)d
Now, let us verify the following simple properties of an A.P
- If a constant is added to each term of an A.P the resulting sequence is also an A.P
- If a constant is subtracted from each term of an A.P the resulting sequence is also an A.P
- If each term of an A.P is multiplied by a constant, then the resulting sequence is also an A.P
- If each term of an A.P is divided by a non-zero constant then the resulting sequence is also known as A.P.
Here, a = FIRST TERM
L - LAST TERM ; D = COMMON DIFFERENCE ; N = THE NUMBER OF TERMS
Sn = the sum to n terms os A.P
→ Now, let a , a + d , a + 2d,......, a + (n-1)d be an A.P. Then l = a + (n-1) d
Sn = n/2 [2a + (n-1)d
→ It can be also written as Sn = n/2[a+l]
I hope you have understood what is a Sequence.