Question

Which one of the following is not true?
A) A sequence is a real valued function defined on $\mathbb{N}$.
B) Every function represents a sequence.
C) A sequence may have infinitely many terms.
D) A sequence may have a finite number of terms.

Answer 1

Hint:
Among the four statements given, one is false. It can be identified from the definition of the sequence. Number of terms of a sequence is not fixed. It can be finite or infinite.

Complete step by step solution:
We are given four statements about sequences.
We have to find out which is wrong among them.
The sequence can be defined as a real valued function with domain, the set of natural numbers. It means, corresponding to every natural number we have a term in the sequence which is a real number.
So we can see option A is a true statement.
Now a sequence may have a finite or infinite number of terms.
This makes option C and option D true.
Now consider option B: Every function represents as a sequence.
This statement is wrong.
Every sequence is a function by definition.
But every function need not be a sequence.
So the statement that is not true is B.

Therefore the answer is option B.

Note:
When we say every sequence is a function, what we mean is that corresponding to every element in the domain, that is a set of natural numbers, there is a unique real number associated. In this sense, a sequence can be considered as the range of a function. For example for the sequence of even numbers, the natural number one has the image two, two has the image four and so on.
Also the infinite terms of the sequence, if exists, need not be distinct.