Question Video: Finding the Coefficient of a Certain Term in a Binomial Expansion Mathematics

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.