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Updated: Nov 27, 2022

There are several ways to find the equation for a quadratic sequence - I have found the following to be the best.

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.

Until the next time... as always, comments and thoughts are welcome.