Chapter 4.6, Problem 50E

To determine

To verify: The claim of the friend regarding range of a function representing an arithmetic sequence.

Answer to Problem 50E

The claim made by the friend is incorrect.

Explanation of Solution

Given information: According to the friend, the range of a function representing an arithmetic sequence contains only positive numbers or only negative numbers.

The claim made by the friend is incorrect.

Consider the case when an arithmetic sequence starts with a positive term and then starts giving negative values, for example, the first term of the arithmetic sequence $2,1,0,-1,-2,\mathrm{....}$ (with common difference $-1$ ) is positive and after a certain stage, the terms of the sequence become negative.

Similarly, consider the case when an arithmetic sequence starts with a negative term and then starts giving positive values, for example, the first term of the arithmetic sequence $-2,-1,0,1,2,\mathrm{....}$ (with common difference $1$ ) is a negative and after a certain stage, the terms of the sequence become positive.

So the above examples contradict the friend’s claim that the range of a function representing an arithmetic sequence contains only positive numbers or only negative numbers.

Thus, the friend is incorrect.