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How to Handle Quadratic Sequences

  • Sep 24, 2021
  • 2 min read

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.


Quadratic sequences take the form:

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

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Which simplifies to…

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

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

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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...

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


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


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


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


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