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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.
2a = 8
So a = 4
Substitute this value for a in 3a + b = 34
3(4) + b = 34
So b = 22
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.
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