Video: Pack 3 β€’ Paper 3 β€’ Question 22

Pack 3 β€’ Paper 3 β€’ Question 22


Video Transcript

The table shows the first five terms of a sequence. Negative eight, negative four, two, 10, 20. Given that the sequence is quadratic, find an expression in terms of 𝑛 for the 𝑛th term of this sequence.

We are told that the sequence is quadratic. That means for the 𝑛th term, its highest power of 𝑛 is two. The general form for the 𝑛th term of a quadratic sequence is given as π‘Žπ‘› squared plus 𝑏𝑛 plus 𝑐, where π‘Ž, 𝑏, and 𝑐 are constants. The first step in finding the 𝑛th term for any linear or quadratic sequence is to find the difference between the terms.

To get from negative eight to negative four, we add four. To get from negative four to two, we add six. From two to 10, we add eight. And from 10 to 20, we add 10. Notice how the difference changes each time. If this was constant, in other words, if the first difference was the same, this would be a linear sequence. However, since the difference is changing each time, that’s a strong indication that it’s a quadratic sequence.

In fact, a quadratic sequence has a common second difference. In this case, the second difference is plus two each time. Notice how this difference remains the same every time, indicating that it is definitely a quadratic sequence. In fact, this number helps us to find the coefficient of 𝑛 squared, represented by the letter π‘Ž in our general form.

To find this number, we halve the second difference. Half of two is one. So the coefficient of 𝑛 squared is one. Remember if the coefficient is one. We don’t actually need to write the number one and we can just write 𝑛 squared as the first part of our 𝑛th term rule.

The next step is to work out the sequence of the term we just found. In this case, the first part of our 𝑛th term is 𝑛 squared. So we’re going to work out the sequence 𝑛 squared. To find the value of the first term, we substitute one for 𝑛. One squared is one. So the first time of our sequence is one. To find the second term, we substitute two for 𝑛. That gives us two squared. And two squared is four. The third term is found by substituting three for 𝑛, giving us three squared which is nine. And the fourth term is given by four squared, which is 16.

Notice that these are simply the square numbers. So we can continue this sequence really easily. The next square numbers are 25, 36, 49, and so on. It is usually enough just to work out the first four terms. But since we know the first five, we’ll stick with them.

Now we want to find out what’s different between the sequence we just generated and the sequence given to us in the question. To do this, we subtract the terms in the bottom sequence from the terms in the top or the original sequence from the question. Negative eight minus one is negative nine, negative four minus four is negative eight, two minus nine is negative seven, 10 minus 16 is negative six, and 20 minus 25 is negative five.

Notice that we’ve actually created another sequence. This time it has a common first difference. So it’s a linear sequence. The first difference is plus one and that gives us the coefficient of 𝑛 in our sequence. It tells us that the value of 𝑏 is one. So the next part of the expression in our 𝑛th term is 𝑛 squared plus one 𝑛 or just 𝑛 squared plus 𝑛.

Finally, we list the sequence one 𝑛 out. We do this by substituting one for 𝑛 to give us one, then two that gives us two, then three that gives us three, which is simply one, two, three, four, five. But what can we do to get from this sequence to the one that we generated earlier? Well, each time, we subtract 10. That means the value of 𝑐 in the 𝑛th term rule for this sequence is negative 10.

In fact, there’s a really easy way to find this number in a linear sequence. What we do is we count back to the zeroth term β€” that is the term that will come in front of the first term. Here, the sequence is going up in ones. So to go back to the number before negative nine, we would do the opposite and go down one, which takes us to negative 10. Remember this only worked for linear sequences, but it can save you a little bit of time.

So currently, we have an expression for the 𝑛th term of this sequence. And what we can do is just check our answer by substituting a couple of values for 𝑛 back into this 𝑛th term. Let’s use 𝑛 equals one to find the first term and 𝑛 equals two for the second. Remember since we’re given the first five terms of the sequence in the question itself, we could check this with any value of 𝑛 between one and five.

When 𝑛 is one, it becomes one squared plus one minus 10, which is negative eight. That value matches the first term in our sequence. And when 𝑛 is two, it becomes two squared plus two minus 10, which is negative four. Once again, this matches the second term in the sequence we were given.

This shows us that we have correctly calculated the 𝑛th term rule of this sequence to be 𝑛 squared plus 𝑛 minus 10.

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