Question

The sequence 6, 12, 24, 48… is a
(a) Geometric Series
(b) Arithmetic Series
(c) Geometric Progression
(d) Harmonic Sequence

Answer 1

Hint: First of all, try to recollect the difference between the terms sequence, series, and progression. Now examine the given sequence and check the general form of its nth term and then check to which category it belongs.

In this question, we have to tell the type of sequence 6, 12, 24, 48….

Before proceeding with the question, let us know about the following terms.

1) Sequence: Sequences are a set of numbers, which are arranged according to any specific rule. The set of numbers should have a definite, logical rule according to which they are arranged. It need not be a mathematical formula, but it should be logical. For example, the following is a sequence of numbers, because they are arranged according to a definite rule:

[5, 7, 9, 11, 13, 15] Rule: nth term = 2n + 3

The following is also a sequence of numbers, as they too have a logical rule:

[4, 6, 8, 9, 10, 12] Rule: Composite numbers

2) Series: A series is a sequence of numbers that are added by + signs. The word ‘series’ is said to represent the sum of the numbers, and not the sum itself. For example, the series associated with the sequence of the above composite numbers is:

[4 + 6 + 8 + 9 + 10 + 12]

3) Progression: Progressions are yet another type of number sets that are arranged according to some definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule like ‘a group of composite numbers’, which does not have a formula associated with it. Therefore, we can say that the above given sequence of composite numbers is not in progression while the sequence of numbers whose nth term 2n + 3 is in progression as well.

Now let us consider the sequence given in the question.

Sequence: 6, 12, 24, 48….

We can also write the above sequence as:

\[6,6\times \left( 2 \right),6\times {{\left( 2 \right)}^{2}},6\times {{\left( 2 \right)}^{3}}....\]

We know that in general, we write a Geometric Sequence like this:

\[\left\{ a,ar,a{{r}^{2}},a{{r}^{3}},.... \right\}\]

Hence the sequence given in the question is a geometric sequence with a = 6 and r = 2.
We know that the general term of a geometric sequence is \[a{{\left( r \right)}^{n-1}}\].

Note: Students get confused between Arithmetic Progression and Geometric progression, it's important to know that in Arithmetic Progression the difference between two numbers is taken and in Geometric progression the ratio between two numbers is taken.