What if the sequence is increasing and begins on a negative value? Or what if the sequence is decreasing and begins on a positive value?
No
Practice makes perfect
Let's consider the given statement. First of all, there are a few things that the friend understood correctly.
The domain is only positive integers. This is because the domain represents the position of the term within the sequence.
The outputs will either constantly increase or constantly decrease. This is because of the common difference.However, these things do not imply that the range will be exclusively positive or exclusively negative. Let's look at an example with an increasing sequence that has a first term that is negative.
a_n= -5+(n-1) 2
In the equation above, the first term is -5 and the common difference is 2. Let's list the first five terms of the sequence.
-5+2 → -3+2 → -1+2 → 1+2 → 3
We can see that the sequence begins with a negative value but ends up with positive values. Therefore, the friend is incorrect.
Extra
Decreasing Sequence With First Term Positive
We can also show that the friend is incorrect using a decreasing sequence that has a positive first term. a_n= 5+(n-1)( - 2)
Let's calculate the first five terms of this sequence.
5-2 → 3-2 → 1-2 → - 1-2 → - 3
Thus, it is possible to start with a positive value and end up with negative values.
