Question

An arithmetic progression with a finite number of terms is called as
(a) A finite arithmetic progression
(b) A finite or an infinite arithmetic progression
(c) An infinite arithmetic progression
(d) A fixed arithmetic progression

Answer 1

Hint: We solve this problem by using the definition of an arithmetic progression.
The sequence of terms formed by either adding or subtracting the fixed number from the first term is called an arithmetic progression. The general representation of a arithmetic progression is given as
\[a,a+d,a+2d,.......\]
Here \['a'\] is called the first term and \['d'\] is called the common difference.
From this definition we find the finite arithmetic progression to check whether the finite A.P is called finite or fixed A.P

Complete step-by-step answer:
We are given the arithmetic progression with a finite number of terms.
We know that the sequence of terms formed by either adding or subtracting the fixed number from the first term is called a arithmetic progression. The general representation of a arithmetic progression is given as
\[a,a+d,a+2d,.......\]
Here \['a'\] is called the first term and \['d'\] is called the common difference.
Let us assume the A.P of finite terms as
\[a,a+d,a+2d,........,{{a}_{n}}\]
Here the term \['{{a}_{n}}'\] is called the last term or \[{{n}^{th}}\] term which can be defined.
Here, we assumed that the A.P is finite which means that the term \['{{a}_{n}}'\] can be defined.
Here, we can see that the term \['{{a}_{n}}'\] is not fixed because we cannot fix the term \['{{a}_{n}}'\] as it depends on different A.P
Let us take some examples of A.P that have finite terms.
(1) 1, 2, 3, 4, 5
(2) 10, 20, 30, 40, 50, 60, 70, 80, 90, 100
(3) 12, 24, 36, 48
Here, we can see that in all examples the terms are different and are finite.
So, we can conclude that the A.P having finite number of terms is called as a finite arithmetic progression

So, the correct answer is “Option A”.

Note: Students may do mistake and take the A.P of finite numbers as fixed A.P
Whatever arithmetic progression we take it cannot be called as fixed A.P.
Sometimes students may take the A.P with a fixed number of terms as a fixed arithmetic progression.
Let us take the A.P of fixed number of terms as follows
(1) 1, 2, 3
(2) 10, 20, 30
(3) 12, 24, 36
Here, we can see that all the A.P have a fixed number of terms of 3 but all the terms are not equal.
So, we can say that an arithmetic progression can never be fixed A.P.