Lesson video

Hi, I'm Miss Davies.

In this lesson, we're going to be looking at proofs with consecutive numbers.

How can we represent consecutive numbers algebraically? If n is an integer, that means the following integer is n add one, then n add two, then n add three, and so on.

Consecutive even numbers are represented with 2n.

The next even number is two more than this, which is 2n add two.

Then, add on another two to give 2n add four and another two to give 2n add six.

Odd numbers are represented with 2n add one or subtract one.

The next odd number after 2n subtract one is two after it.

If we add two onto 2n subtract one, this gives us 2n add one.

Another two to the next odd number is 2n add three, and then the next odd number is 2n add five.

We are going to prove that the sum of two consecutive odd numbers is a multiple of four.

Our first odd number that we're going to use is 2x add one.

The next odd number is 2x add three.

If we add these together, it gives a result of 4x add four.

We can then take four out as a factor, meaning that this is a multiple of four.

We're going to prove that the difference between the squares of two consecutive integers is equal to the sum of the two integers.

Two consecutive integers can be written as x and x add one.

If we're finding the difference between the squares of these two integers, we're going to do x add one squared subtract x squared.

This gives x squared add 2x add one subtract x squared.

This simplifies to give 2x add one, which can be written as x add x add one, which is the sum of the two initial integers.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

You might've used different expressions to represent your consecutive numbers, but you still should have been able to take a factor of the given number.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

No matter what expression you have used to represent your consecutive numbers, you should still be able to take five as a factor.

In this next example, we're going to prove that the product of three consecutive odd numbers is not a multiple of four.

Three consecutive odd numbers that we're going to use are 2x add one, 2x add three, and 2x add five.

To find the product of these three consecutive odd numbers, we're going to multiply them together.

This gives 8x cubed add 16x squared add 46x add 15.

We can take four out as a factor of the first two terms, but we are unable to take a factor of four from the final two terms. Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

Six multiplied by x add one add three cannot be a multiple of six.

It is three more than a multiple of six.

That's all for this lesson.

Thanks for watching.