Question

If a series consists only of a finite number of terms it is called a ____________.
(a) Infinite Series
(b) Finite Series
(c) Real Number
(d) Geometric Series

Answer 1

Hint: To solve the above question, we will find out what series is. Then after knowing what a series is, we will check each option one by one. While checking each option, we will find out what these series are. If the series will have a limited number of terms, i.e. the series ends after a certain term, then that will contain finite terms and the option containing this type of series will be the correct option.

Complete step-by-step answer:
Before we solve this question, we must know what a series is. A series is simply the sum of the various numbers or elements of a sequence. A series is represented by
\[S={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....+{{a}_{n}}\]
\[\Rightarrow S=\sum\limits_{i=i}^{n}{{{a}_{i}}}\]
where \[\sum \] is the summation sign and n is the number of terms in that series. When the value of n is finite, i.e. the number of terms is limited, then that series contains a finite number of terms. When \[n\to \infty ,\] then the series does not contain a finite number of terms. Now, we will check each option one by one.
Option (a): Infinite series
When the ‘n’ in the series \[S=\sum\limits_{i=i}^{n}{{{a}_{i}}}\] tends to infinite, the series becomes infinite series and the number of terms in this series is infinite.
Option (b): Finite series
When the ‘n’ in the series \[S=\sum\limits_{i=i}^{n}{{{a}_{i}}}\] is a finite number then the series becomes finite series and the number of terms in this series is finite.
Option (c): Real Number
Real numbers are a combination of rational and irrational numbers. The range of real numbers is \[R\in \left( -\infty ,\infty \right).\] Thus, the real numbers contain an infinite number of terms.
Option (d): Geometric Series
The series in which the ratio between the two consecutive terms is constant. It is represented by
\[S=a+ar+a{{r}^{2}}+.....a{{r}^{n}}\]
where \[n\to \infty .\] It also contains an infinite number of terms.
Hence, option (b) is the right answer.

Note: One may get confused that the infinite series have the infinite sum and finite series have the infinite sum. This is not the case. Infinite series can also have a finite sum. For example,
\[S=1+\dfrac{1}{2}+\dfrac{1}{4}+.....\]
The sum of the above series is 2. This the infinite series and the finite series are called so because of their number of terms and not on the basis of their sums.