Question

A sequence of numbers in which each term is related to its predecessor by same law is called
(a) arithmetic series
(b) progression
(c) geometric series
(d) none of these

Answer 1

Hint: An arithmetic series is the sum elements of the sequence. In arithmetic series, we need not be concerned about the order of the elements. Progression or sequence is a set of elements that follow a pattern. Here, the order of the elements is important. Progression can be Arithmetic Progression, Geometric Progression and Harmonic Progression. A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms.

Complete step-by-step answer:
Let us first recollect what an arithmetic series is. An arithmetic series is the sum elements of the sequence. In arithmetic series, we need not be concerned about the order of the elements. For example, let us consider $1+3+5+7+..$ . This is an arithmetic series because it is the sum of elements. $5+7+3+1+...$ is also an arithmetic series. Here, we can conclude that the order of elements is not important.
Now, let us understand progression. Progression or sequence is a set of elements that follow a pattern. Here, the order of the elements is important. Progression can be Arithmetic Progression, Geometric Progression and Harmonic Progression. Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. For example, let us consider a sequence 1, 3, 5, 7, … . Here, we can see that $3-1=2,5-3=2,7-5=2,..$ , that is the difference of two consecutive numbers is a constant value 2. Hence, this sequence is an AP. Geometric progression (GP) is a sequence of numbers such that the quotient of any two successive elements of the sequence is a constant. For example, let us consider a sequence $a+ar+a{{r}^{2}}+a{{r}^{3}}+...$ . Here, we can see that $\dfrac{ar}{a}=r,\dfrac{a{{r}^{2}}}{ar}=r,\dfrac{a{{r}^{3}}}{a{{r}^{2}}}=r,...$ . Hence, the constant ratio is r. Harmonic progression (HP), is sequence of numbers such that their reciprocals form an arithmetic progression. For example, we can call the progression $1,\dfrac{1}{3},\dfrac{1}{5},\dfrac{1}{7},...$ as HM, since they are the reciprocal of AP 1, 3, 5, 7, … .
Let us look into geometric series. A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. We can say that geometric series is the sum of elements in the GP. For example, let us consider a series $a{{r}^{2}}+a+a{{r}^{2}}+...$ . This series is the sum of the elements of a GP. Here also, the order of elements is not important.

From the above definitions, we can conclude that a sequence of numbers in which each term is related to its predecessor by the same law is called a progression. This is because in the other two, each term is not related to its predecessor by the same law or constant because the order of elements is not important.

So, the correct answer is “Option b”.

Note: We can directly find the correct option because arithmetic series and geometric series are series where the order of elements is not concerned.Hence, these series will not have each of the term related to its predecessor by the same law. Let us consider an arithmetic series $5+7+3+1+...$ . Here, we cannot find the common difference $\left( 7-5=2,3-7=-4 \right)$ . Similarly, for geometric series, we cannot find the constant ratio. For example, for a geometric series $a{{r}^{2}}+a+a{{r}^{2}}+...$ , there is no common ratio.