Video Transcript
Determine the coefficient of 𝑎
squared in the expansion of 𝑎 over 11 plus 11 over six 𝑎 to the power of 12.
In order to actually find the
coefficient of 𝑎 squared in our expansion, what we’re gonna use something called
the general term formula. And what the general term formula
tells us is that any term is equal to 𝑛 choose 𝑟 multiplied by 𝑎 to the power of
𝑟 multiplied by 𝑏 to the power of 𝑛 minus 𝑟. And this is when we have our
expansion in the form 𝑎 plus 𝑏 to the power of 𝑛.
So now, we actually have this
general term formula and we know what each part is. What we want to actually do is see
how we can use this to help us find our coefficient of 𝑎 squared. Well, first of all, what I’m gonna
do is actually write down what we know. So we’re gonna write down 𝑎. Well, this is 𝑎 over 11 because as
you can see from our general term formula 𝑎 is the first term inside our
parentheses. And 𝑏 is equal to positive 11 over
six 𝑎. And it’s important here to be
careful and remember the signs because if we had negative 11 over six 𝑎, we’d have
to write that down because it actually makes a difference to the answer when we’re
working through.
Okay, now, we’re gonna move on to
𝑛. Well, 𝑛 is equal to 12 because
actually that’s the exponent of our parenthesis. Then, we’ll move on to 𝑟. And actually 𝑟- we don’t know what
𝑟 is and that’s what we need to find out because this is gonna help us to determine
the coefficient of 𝑎 squared.
Okay, so now we have these. Let’s substitute them back into our
formula. So what we get is our term is equal
to 12 choose 𝑟 multiplied by 𝑎 over 11 to the power of 𝑟 multiplied by 11 over
six 𝑎 to the power of 12 minus 𝑟. So now what I’ve done is actually
separated out our terms. So we can actually see our 𝑎 terms
on their own. So we’ve got 12 choose 𝑟
multiplied by one over 11 to the power of 𝑟 multiplied by 𝑎 to the power of 𝑟
multiplied by 11 over six to the power of 12 minus 𝑟 and then multiplied by 𝑎 to
the power of negative 12 minus 𝑟.
And I got our final term 𝑎 to the
power of negative 12 minus 𝑟 using exponent rule. And that rule tells us that if we
got one over 𝑎 to the power of 𝑏, then this is equal to 𝑎 to the power of
negative 𝑏. Okay, great, so we’ve now gotta
like this. What do we do next? Well, if we look back at the
question, we’re interested in the term that actually has 𝑎 squared. So therefore, what we are
interested in is these terms here: our 𝑎 to the power of 𝑟 and 𝑎 to the power of
negative 12 minus 𝑟. Because we can say that in the term
that we’re looking for, 𝑎 to the power of 𝑟 multiplied by 𝑎 to the power of
negative 12 minus 𝑟 must be equal to 𝑎 squared.
Now, where we set up an equation,
we can see that our bases are the same. So we’ve got 𝑎 is the base for
each of our terms. So therefore, we can actually
equate our exponents. So then, we’ve got 𝑟 plus negative
12 minus 𝑟 is equal to two. So we actually got this using
another exponent rule, which tells us if we’ve got 𝑎 to the power of 𝑏 multiplied
by 𝑎 to the power of 𝑐, then they both got the same bases and what we do is we
just add the exponents. So in this case, we added 𝑟 and
negative 12 minus 𝑟. So then we simplify and we get 𝑟
minus 12 plus 𝑟 equals two. And then we add 12 to each
side. So we get two 𝑟 is equal to
14. And then, we divide by two. So we get that 𝑟 is equal to
seven.
Okay, great, we’ve now found our
𝑟. So let’s take this and actually
substitute it back into our formula to find out what our terms going to be. So we’re gonna have that the 8th
term is equal to 12 choose seven multiplied by one over 11 to the power of seven
multiplied by 𝑎 to the power of seven multiplied by 11 over six to the power of 12
minus seven which is five and then multiplied by 𝑎 to the power of negative 12
minus seven which is negative five.
Okay, great, so now what we need to
do? When we look back at the question,
and the question asks us to actually find the coefficient of 𝑎 squared. So we know that this term does
involve the 𝑎 squared term because 𝑎 to the power of seven multiplied by 𝑎 to the
power of negative five gives us 𝑎 squared. So what we need to do to actually
find the coefficient is actually to multiply together our other components. So we’ve got 12 choose seven
multiplied by one over 11 to the power of seven multiplied by 11 over six to the
power of five which is equal to one over 1188.
So therefore, we can say that the
coefficient of 𝑎 squared in the expansion of 𝑎 over 11 plus 11 over six 𝑎 to the
power of 12 is one over 1188.
