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Sequences, Series and Progression is considered one of the most asked concept.
25 Questions around this concept.
Statement 1:The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + ...... + (361 + 380 +400) is 8000.
Statement 2: for any natural number n.
Sequences:
A sequence is formed when terms are written in order such that they follow a particular pattern
Eg, 1, 2, 3, 4, 5,.....
1, 4, 9, 16, ......
1/3, 1/4, 1/5, 1/6, ......
Finite and Infinite Sequences
A sequence may have an infinite number of terms or a finite number of terms.
If a sequence has three dots at the end, then it indicates the list never ends. It has infinite numbers of terms. Then, the sequence is said to be an infinite sequence.
Eg, 2, 4, 6, 8, 10, …
If sequence has only a finite number of terms, then the sequence is called a finite sequence.
Eg, 2, 4, 6, 8
5, 8, 11, 14, ......., 65
nth term
In sequences nth term is usually denoted by an or tn or Tn
So, in sequence 2, 4, 6, 8, ...
a1= 2, a2= 4, a3= 6, and so on
We can write it in compact form as an = 2n
an or nth term is also called General term of the sequence.
So a sequence can be written as a1, a2, a3, ........
Conversely, if the general term of a sequence is given, we can find any term of that sequence.
Eg, If Tn = 2n , then fourth term can be obtained by putting n = 4
So, T4 = 24 = 16
Series:
If we add or subtract all the terms of a sequence we will get an expression, which is called a series. It is denoted by Sn.
This is the formula for finding general term of a sequence if the sum of n terms is given.
Progression:
If the terms of a sequence follow some pattern that can be defined by an explicit formula in n, then the sequence is called a progression.
Eg. 3, 9, 27, 81…
Here, general term can be written explicitly in terms of n, which is 3n
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