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Maths Problem of the Week

Next Three Terms of a Quadratic Sequence


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Find the next three terms in this sequence

14, 24, 36, 50, 66, ..., ..., ...,

-oOo-

Solution

... to follow ...

As with any sequence we are given first let's look to see if it is not one we know. It's not one I know, so let's find the first difference between adjacent terms.

n 1 2 3 4 5
nth term
(Un)
14 24 36 50 66
First difference
(Un+1-Un)
10 12 14 16

This first difference is an arithmetic sequence with a common difference of 2, which enables us to work out the next three first differences:

n 1 2 3 4 5 6 7 8
nth term
(Un)
14 24 36 50 66 ? ? ?
First difference
(Un+1-Un)
10 12 14 16 18 20 22

Adding these first differences in term to the previous term in the sequence we get:

n 1 2 3 4 5 6 7 8
nth term
(Un)
14 24 36 50 66 84 104 126
First difference
(Un+1-Un)
10 12 14 16 18 20 22

So the sequence is 14, 24, 36, 50, 66, 84, 104, 126,…

We can go further and find any term in the sequence. The first difference is an arithmetic sequence with a common difference of 2, implying that the original sequence is a quadratic sequence of the form Un=an2+bn+c, where a is the common difference divided by 2, ∴ a=1.

n 1 2 3 4 5 6 7 8
nth term
(Un)
14 24 36 50 66 84 104 126
First difference
(Un+1-Un)
10 12 14 16 18 20 22
an2 1 4 9 16 25 36 49 64
Un-an2 13 20 27 34 41 48 55 62

But Un-an2=bn+c, so the last line is an arithmetic sequence with common difference b=7 and zeroth term c=6. Therefore, the nth term of the sequence is given by Un=n2+7n+6.

-oOo-





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