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 𝑎𝑛 squared plus 𝑏𝑛 plus 𝑐, where 𝑎, 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
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 𝑎 in our general
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
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 — 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