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CCSS.Math: , ,

- [Instructor] The expression
five times two to the T gives the number of leaves in a plant as a function of the number
of weeks since it was planted. What does two represent
in this expression? So pause this video and see
if you can figure it out on your own. All right, so let's look at
the expression right over here. We could write it as defining a function. So we could say leaves
as a function of time is equal to five times two to the T power and so we could try this out a little bit. If we say well what is L of zero? That would be T equals zero. That's when we are at zero
weeks after it was planted. So this is right when it was planted. Well that's five times two to the zero, which is just two to the zero's just one. So it's equal to five and so when you see an
exponential expression or an exponential function like this, that is why this number out here is often referred to
as your initial value. Initial. Initial value. And so let's explore
this a little bit more. What is L of one? What happens after one week? Well that's gonna be five
times two to the first power or five times two. So going from when it was
planted to the first week, we are multiplying by two. The number of leaves doubles. Well what happens after two weeks, the number of leaves after two weeks? Well that's gonna be five
times two to the second power. Well that's the number that
you had in the first week times two. So it looks like every
week we are doubling, we are multiplying by two and that's why this
number right over here, which is what the question
is about, the two, this is often referred
to as the common ratio. Common ratio. Because between any two successive weeks, the ratio between say week
two and week one is two. Week two is double week one. Week one is double week zero. So let's see which of these choices actually match up to that. There were initially
two leaves in the plant. Well we know that there weren't
two leaves on the plant. Our initial value was five. So let me cross that one out. The number of leaves is
multiplied by two each week. Well that's exactly
what we just described. So I like that choice. Let's look at the last
one just for good measure. The plant was planted two weeks ago. Well no, they don't tell
us anything about that. This is a general expression for T weeks after it was planted. So they're not saying when
it was actually planted. So we could rule that out and we feel good about that second choice.