JEE Advanced Important Questions of Arithmetic and Geometric Progressions
FAQs on JEE Advanced Arithmetic and Geometric Progressions Important Questions
1. What is a sequence and what is a series?
When students study arithmetic, they come over these topics of sequence and series. A collection of digits in which numbers could repeat are altogether known as a sequence. Series is the sum of all the elements in the sequence. Numbers in sequence are arranged in a particular pattern and numbers in a series are mostly generalized yet, there is some relationship between the numbers. They may look very similar but when you practice more problems, you will know the difference between them. The term length of the sequence represents the number of elements in it. The major difference is in a sequence the numbers have a chance to get repeated. There are different types of sequences like arithmetic sequence, geometric sequence, harmonic sequence, etc.
2. What is geometric progression and its general term?
Geometric progression is a subtype of the sequence where every next number is a product of the preceding number with a constant value. It is also known as a geometric sequence. For example, 3,6,12,24…...in this series, the common ratio between two consecutive numbers is 2. The general form of a geometric progression is \[ a, ar^{2}, ar^{3}, ... ar^{n}\]. There are two types of geometric progressions namely finite geometric progression and infinite geometric progression. Here, a denotes the first term, r is the common ratio and arn is the nth term. Read the above PDF to know in detail about the geometric progression.
3. What is an arithmetic progression and what is its general term?
Arithmetic progression is a subtype of sequences in which the difference between a term and its previous term is the same for the entire series. For example, if the sequence of the numbers is 2,4,6.8,... the difference between two consecutive numbers in the series is 2. Hence, we can say that the series is in arithmetic progression. In a progression, the common difference is denoted by ‘d’. The nth term is denoted by ‘\[a^{n}\]’.
Nth term of AP: \[a^{n} = a + (n - 1) \times d\].
Similarly, sum of N terms is defined by \[ \frac{n}{2} (2a + (n - 1) \times d)\]
The Sum of AP when the last term is given is defined by n/2 (first term + last term)
General form of AP is a, a + d, a + 2d, a + 3d, . . .
4. What is a Harmonic progression and what is its general term?
Harmonic progression is a sub-type of sequence in which real numbers are arranged in a particular format where their reciprocals of Arithmetic progression does not contain zero. For example p,q,r,s,.... Are in arithmetic progression, then their harmonic progression is 1/p,1/q,1/r,1/s,...
The nth term of the Harmonic Progression \[(H.P) = \frac{1}{(a+(n-1)d)}\]
Here, a is the first term of the arithn=metic progression
d is the common difference
n is the number of terms in arithmetic progression
5. What is the Fibonacci sequence?
A Fibonacci sequence is a set of numbers in which the sequence starts with 0 and 1 and the next element is obtained by adding two previous elements. General expression of the Fibonacci sequence is \[F^{0} = 0\] and \[F^{1} = 1\] and \[F^{n} = F^{n-1} + F^{n-2}...\] It is an infinite sequence. This is also known as ‘nature’s secret code’, it can be found in sunflower petals, cauliflowers, seashells, etc. It is represented by the spiral format. The spiral starts with a rectangle then it is partitioned into squares and then they are further partitioned. It is used in grouping numbers, coding, and also in making music.