Does linear growth have to add the exact same amount, so a constant amount each time such as 500, 500, 500, etc, or does it just have to be close, such as 125, 137, 145, 160, etc?
Linear growth does not necessarily have to add the exact same amount each time. It can be a constant amount added at each step, like in your first example (500, 500, 500), which is known as arithmetic progression. However, it can also be close but not the same amount added each time, like in your second example (125, 137, 145, 160), which is known as a quasi-linear or approximate linear growth.
To determine if a sequence is a linear growth, you can use the formula for an arithmetic progression, which is:
nth term = a + (n - 1)d
Where:
- nth term is the term in the sequence you want to find,
- a is the first term in the sequence,
- n is the position of the term in the sequence (starting from 1),
- d is the common difference between each term.
If the formula holds true for the given sequence, it indicates linear growth.
In your first example, the common difference is 500, and if you substitute the values into the formula, it will correctly calculate any term in the sequence.
In your second example, there is no constant difference between each term. However, if you calculate the differences between consecutive terms, you may notice that they are not the same but are approximately increasing or decreasing. This indicates a quasi-linear growth, where the increase or decrease in each step is similar, but not exactly the same.
Remember, linear growth can have either a constant amount added each time or a close approximation, depending on the specific values in the sequence.