# Question #cc09c

##### 1 Answer
Jan 30, 2018

The next number is 25.

The formula is

$y = 200 {\left(\frac{1}{2}\right)}^{x} , x \in \mathbb{N}$

#### Explanation:

This is an exponential decay or half-life type of question. The number halves each time, so the number after 50 is 25.

The general equation for exponential decay can be expressed as

$y = a {b}^{x}$

where $a$ is the initial amount, and $b$ is the base. In this case we want a base of 1/2, because the number is halving each time.

If the number sequence was 200, 50, 12.5, etc. we would use a base of 1/4 because the number is being quartered each time. The initial amount can also be any number, but in this case $a = 200$. Plugging these into the equation, we get

$y = 200 {\left(\frac{1}{2}\right)}^{x} , x \in \mathbb{N}$

$\mathbb{N}$ are the natural numbers (positive integers including zero).

Notice that we had to restrict the values that $x$ can take because there are a discrete number of values that we want. For example, $x$ can't be 1.5, because that would give a number that is not one of our answers.

The graph looks like this

The important thing here is that $x$ can only take on whole number values, which is why we restricted the domain to natural numbers. The line drawn through the points is only there because I don't know how to turn it off.