• Revision:Nth Term

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The nth is an expression in terms of n which is to be found to reproduce a series of numbers. I.e. the nth term of the sequence:

9, 11, 13, 15, 17, 19

is 2n + 7.

n 1 2 3 4 5 6
2n + 7 9 11 13 15 17 19

Linear nth Terms

For any sequence such as 9, 11, 13, 15, 17, 19, where there is a common difference (i.e. 2), you can always find the nth term using the formula:

n^{th}\, term = dn + (a - d)
a = The first term in the sequence
d = The common difference between the terms

I.e. n^{th}\, term = 2 \times n + (9 - 2) = 2n + 7

Quadratic nth Terms

For any sequence such as 3, 3, 5, 9, 15, 23, where the second difference is the same (i.e. the first differences between each term are 0, 2, 4, 6, 8 and the second difference is 2), you can always find the nth term using the formula:

n^{th}\, term = a + (n - 1)d + \frac{1}{2}(n-1)(n-2)C
a = The first tern
d = The difference between the first and second term
C = The common second difference.

I.e. n^{th}\, term = 3 + (n-1) \times 0 + \frac{1}{2}(n-1)(n-2)\times 2 = 3 + (n-1)(n-2) = n^2 - 3n + 5

Method of Finite Differences

An alternative to learning the formulae is the method of finite differences. You start by finding the difference between each term and, using the differences as your new sequence, repeat the process until a common difference is found.

Image:Nthterm1.png

In this example the sequence has a second-order common difference of two. The next step is to compare the original sequence with \frac{d}{o!}n^o where d is the common difference and o is the order of the difference. Then repeat the process.

Image:Nthterm2.png

From this we can see the nth term is: n^2 - 3n + 5.

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