Quadratic Sequences

Quadratic sequences take the form:

If we substitute the first term (1) into the general form, we get…

Which simplifies to…

This is shown in the table below immediately beneath the term’s value.

Here we have a sequence of numbers. I have included the differences between the numbers in the sequence.

Notice that the 2nd difference is constant across the sequence- a sure sign that you have a quadratic sequence.

Finding the next number in the sequence is easy - add the 1st and second differences together from n=4 to n=5 (58 + 8 = 66) then add this to the value for n=5 (207 + 66 = 273). The value for n=6 is 273.

Let's find out the quadratic equation that gave this sequence of numbers...

From the derived equations in the first table, we can calculate the values for a, b and c.

Sequence = a + b + c = 23

1st difference = 3a + b = 34

2nd difference = 2a = 8

Of course, you can memorise the formulas, but they can still be worked out if you forget.

For a:

2a = 8

So a = 4

Substitute this value for a in 3a + b = 34

3(4) + b = 34

So b = 22

Finally…

a + b + c = 23

4 + 22 + c = 23

c = -3

We can put these values into the quadratic equation and, voila, get...

Make sure you check it with a value for n, for example n = 2... we should get 57, the second number in the sequence.

What if we need to find a value in the sequence, perhaps the number for when n = 20? Just substitute into the equation...

When n = 20, the number in the sequence is 2037.