Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 1 β€’ Question 9

Which of the following expressions can be used to calculate the sum of three consecutive odd numbers?

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Video Transcript

Which of the following expressions can be used to calculate the sum of three consecutive odd numbers? Circle your answer. Three multiplied by two 𝑛 plus two, three multiplied by two 𝑛 plus one, six 𝑛 plus one, or two 𝑛 plus nine.

So when we’re looking at the question, we can see that we want to find the sum of three consecutive odd numbers and then find one of our expressions that can be used to calculate this. Well, first of all, what does three consecutive numbers mean? Well, consecutive numbers are numbers that come directly after each other.

But in particular, we want consecutive odd numbers. So that would mean, for instance, one, three, five or five, seven, nine. So if we consider 𝑛 to be a number, how can we make sure that the number is going to be an odd number? Well, if we think about two 𝑛, we know that two 𝑛 is going to be an even number. That’s because two multiplied by any number is even.

And two examples to show this out: two multiplied by one, so that’s an odd number, gives us an answer of two, which is an even number. And two multiplied by four which is an even number gives us an answer of eight which is again an even number.

So therefore, we can say that two 𝑛 minus one and two 𝑛 plus one must be odd numbers because the numbers either side of an even number are odd. As an example, we have one, two, three.

So two is an even number. And then either side, so if we take one away, we get one which is an odd number. And then, if we add one on, we get three which again is an odd number.

Again, if we look at seven, eight, nine, eight is an even number. Take one away, we get seven which is an odd number. Add one on, we get nine which again is an odd number.

So now, what we need to think about is the sum of three consecutive odd numbers. So if we start with our two 𝑛 minus one, then the next odd number would be two 𝑛 plus one, then the next odd number would be two 𝑛 plus three. And that’s because we’re having a difference of two each time.

And that’s because between each odd number, we would have a difference of two. So between one and three, there is a difference of two. Between seven and nine, there’s a difference of two. And then starting from two 𝑛 plus one, the next two consecutive odd numbers would be two 𝑛 plus three and two 𝑛 plus five.

Now the sum means adding them together. So let’s add first of all two 𝑛 minus one, two 𝑛 plus one, and two 𝑛 plus three. So first of all, we’ve got minus one add one which is zero, then add three which is three. And then, we have two 𝑛 add two 𝑛 which is four 𝑛 add another two 𝑛 which is six 𝑛.

So then, if we look at our next three consecutive odd numbers, we’ve got two 𝑛 plus one add two 𝑛 plus three add two 𝑛 plus five. So first of all, if we do one plus three plus four plus five is nine. And again two 𝑛 add two 𝑛 is four add another two 𝑛 is six 𝑛.

Okay, so do either of these look like the expressions that we’ve got in the question? So as you can see that they don’t fit any of the expressions that we have in our answer. Then, what we can do is factorize.

Always watch out for factorization in this type of question. And we know you can factorize cause if we take a look at each of the answers to the sums, we can see that there are common factors in the terms.

On the left-hand side, we can see that a common factor of six 𝑛 and three is three. And this is the highest common factor. And then inside the bracket, we’re gonna have two 𝑛 plus one. And that’s cause three multiplied by two 𝑛 is six 𝑛 and three multiplied by one or positive one gives us positive three.

And then again, if we look at the right-hand answer to the sum, we can see that three would be the highest common factor. So we take three outside the brackets. And then inside the brackets, we’ll have two 𝑛 plus three. And that’s because three multiplied by two 𝑛 is six 𝑛 and three multiplied by positive three gives us positive nine.

And now, we can see if either of these match our expressions. And yes! We can see that one of them does because we got three multiplied by two 𝑛 plus one which matches our second expression.

And therefore, we can say that the expression three multiplied by two 𝑛 plus one can be used to calculate the sum of three consecutive odd numbers.

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