Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 1 • Question 25

A sequence of numbers is formed by the iterative process 𝑎_(𝑛 + 1) = (2𝑎_(𝑛))² − 2𝑎_(𝑛). (a) Describe the sequence of numbers when 𝑎₁ = 1/2. You must show your working. (b) Describe the sequence of numbers when 𝑎₁ = −1/4. You must show your working. (c) Work out the value of 𝑎₂ when 𝑎₁ = (1/2) − √3.

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Video Transcript

A sequence of numbers is formed by the iterative process 𝑎 𝑛 plus one is equal to two 𝑎 𝑛 all squared minus two 𝑎 𝑛. Part a) Describe the sequence of numbers when 𝑎 one is equal to a half. You must show your working. Part b) Describe the sequence of numbers when 𝑎 one is equal to negative a quarter. You must show your working.

There’s also part c that we’ll come to. So the first thing to look at with this question is this term “iterative process.” So what is an iterative process? Well, an iterative process is a repeated process that is convergent to a particular value. So we’ll see that the sequences we’re gonna have a look at should converge to a particular value.

So to start within part a, what we’re gonna do is substitute in a half as our 𝑎 one value. So therefore, our second term value is gonna be equal to two multiplied by a half all squared because that’s part of our iterative process and then minus two multiplied by a half because again that was what our first term value was, a half. So this is gonna be equal to one squared minus one which will give us a value of zero. So we can say that the second term is gonna be equal to zero.

Now, let’s move on to the third term. Well, to work out the third term using the iterative process, what we’ll now gonna do is put the value from the second term in, so that’s zero. So what we’re gonna get is the third term is gonna be equal to two multiplied by zero all squared minus two multiplied by zero. So therefore, this is gonna be equal to zero minus zero which will give a result of zero.

So what we can see already is that our iterative process when you start with the first term being a half has converged. And it’s converged to a particular number and that is zero. So therefore, we can say that from the second term onwards, all terms are equal to zero. So that’s our description to the sequence and we’ve shown our working. So now we can move on to part b.

So in part b, this time, our first term is equal to negative a quarter. So we’re gonna substitute that in to try and find our second term. So we get two multiplied by negative a quarter all squared minus two multiplied by negative a quarter. So this is gonna give us negative a half all squared and that’s cause two multiplied by negative a quarter is negative a half then plus a half and that’s because we had negative two multiplied by negative a quarter. Negative multiplied by a negative is a positive and two quarters is the same as a half. And this is gonna give us a quarter plus a half and that’s cause negative a half all squared is the same as negative a half multiplied by negative a half.

So one multiplied by one is one, two multiplied by two is four, and a negative multiplied by a negative is a positive. So we got a quarter plus a half which will give us an answer of three quarters. And we got three-quarters because if we had a quarter plus a half, well you’ve got a quarter plus two quarters and that’s because a half is the same as two-quarters. So we add the numerators. It gives us three over four or three-quarters.

So now, we’re gonna to take a look at the third term. Well, if we take a look at the third term, we got two multiplied by three-quarters all squared minus two multiplied by three-quarters. Well, this is gonna give us six over four all squared and that’s cause we have two multiplied by three over four which gives us six over four or six-quarters minus six-quarters because two multiplied by three over four is gonna give us six over four or six-quarters which is gonna give us 36 over 16 minus six over four and that’s cause we square the numerator and denominator.

And then, what we do is we convert our first fraction into quarters. We can do that by dividing the numerator and denominator by four. So we get nine over four minus six over four which will give us three over four or three-quarters which is the same as we got in our second term of the sequence. So once again, we can see that we’ve converged it to a value. And that value is three-quarters.

So we can say that using our iterative process and starting with our first term of negative a quarter, we can say that from the second term onwards all terms are equal to three-quarters or three over four. So now, we’ve solved part a and part b. Let’s move on to part c.

So part c) Work out the value of 𝑎 two or the second term when the first term is equal to a half minus root three.

So what we’re going to do is substitute in our first term is equal to a half minus root three. And when we do that, we’re gonna get the second term is equal to two multiplied by a half minus root three. And then this is all squared minus two multiplied by a half minus root three.

So we’re gonna start by expanding the inside of the first bracket. So we got two multiplied by a half which gives us one. And then we’ve got two multiplied by negative root three. So we’ve got negative two root three and then this is all gonna be squared. And then, we’re gonna have minus one because we got negative two multiplied by a half which gives us minus one. And then, we got plus two root three, being careful not to avoid a common mistake here because many students will put minus two root three. But we’ve got negative two multiplied by negative root three. So that’s why we get positive two root three.

So now, what we need to do is expand the first bracket because we got one minus two root three all squared. So it’s the same as one minus two root three multiplied by one minus two root three. So first of all, we’re gonna get one multiplied by one which just gives us one. Then we’ve got one multiplied by negative two root three. So we get negative two root three. And then negative two root three multiplied by one which gives us another negative two root three.

And then, for the final term, we’re gonna get plus four multiplied by three and we get that because we got negative two root three multiplied by negative two root three. Well, negative two multiplied by negative two is four. And we know that root three multiplied by root three is just gonna be equal to three. And that’s because we got a rule that root 𝑎 multiplied by root 𝑎 is equal to 𝑎.

And the reason that is is because if we think about root three multiplied by root three, well root three multiplied by root three would be root nine and root nine is just three. So then, we get one minus four root three because we’re collecting negative two root three and negative two root three then plus 12. So then, we put this in and we get one minus four root three plus 12 minus one plus two root three.

So we’ve got one and then minus one. So these cancel out. Then, we’ve got positive 12. And then, we’ve got negative four root three plus two root three which is gonna give us negative two root three.

So therefore, we can say that the value of the second term when the first term is equal to a half minus root three is going to be 12 minus two root three.

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