Below is a sequence containing six
terms: six, 13, 26, 45, 70, and 101. Write an expression in terms of 𝑛
for the 𝑛th term of this sequence.
A good place to start is to look
for the first difference of the sequence. Notice how the difference changes
each time. This means this is not an
arithmetic sequence. However, the second difference is
plus six each time. This means it’s a quadratic
sequence, 𝑛 squared.
To find the coefficient of 𝑛
squared, which is the number in front of 𝑛 squared, we halve the second
difference. Half of six is three. So our coefficient of 𝑛 squared is
three. Next, we should write the sequence
three 𝑛 squared out above our original sequence.
When 𝑛 is one, three 𝑛 squared is
three multiplied by one squared, which is three. When 𝑛 is two, three 𝑛 squared is
three multiplied by two squared, which is 12. Remember, we must work out the
index or the power before we multiply. When 𝑛 is three, three 𝑛 squared
is three multiplied by three squared, or 27. And when 𝑛 is four, three 𝑛
squared is three multiplied by four squared, which is 48. We don’t need to write out the
whole sequence. Three or four terms is sufficient
to help us identify a pattern.
Now let’s see what we need to do to
get from our sequence of three 𝑛 squared to the sequence we were given in the
question. For the first term, we add
three. To get from 12 to 13, we add
one. To get from 27 to 26, we subtract
one. And to get from 48 to 45, we
Take a look at this sequence. You might have noticed we’ve
created a linear sequence. This sequence has a common
difference of negative two. So its 𝑛th term starts with
negative two 𝑛. The negative two times table is
negative two, negative four, negative six, and negative eight.
To get from this sequence to ours,
we add five each time. So the 𝑛th term for this new
sequence that we created is negative two 𝑛 plus five. Putting this all together, we get
that the 𝑛th term of the quadratic sequence is three 𝑛 squared minus two 𝑛 plus