The nth Term of a Quadratic
Y10
We have already used the difference method to find a quadratic function from a table.
When looking at sequences always see if it is a simple function before attempting the ‘rote’ methods. The
difference method is useful to discover if a sequence is linear or quadratic. Remember, if the first difference
is constant the sequence is a linear function, if the second difference row is constant then the function is a
quadratic.
Here is another method to find a quadratic function from a quadratic sequence.
Find the nth term of the sequence 3, 12, 25, 42, 63.
3
9
12
25 42 63 ....
13 17 21
4
4
4
1st difference row
2nd difference row is constant.
We know that the nth term of a quadratic sequence is of the form
an2 + bn + c
where a, b and c are numbers.
If we put the general equation and sequence together
For the first term i.e. n = 1,
For the second term i.e. n = 2,
For the third term i.e. n = 3,
a+b+c=3
4a + 2b + c = 12
9a + 3b + c = 25
By subtracting these we can create two simultaneous equations
4a + 2b + c = 12 _
a + b + c = 3
3a + b
= 9
9a + 3b + c = 25 _
4a + 2b + c = 12
5a + b
= 13
These can be solved to give 2a = 4, i.e. a = 2 and hence, by substitution, b and c can be found.
In this case b = 3 and c = -2.
These can now be put back into the general equation for the nth term.
The nth term of the sequence 3, 12, 25, 42, 63... is 2n2 + 3n - 2
For each of the following
a).
find the next two terms,
b).
find the nth term for the quadratic sequences.
See if the sequence is a simple quadratic function before attempting the rote methods.
1). 4). 7). 10). 13). 16). 19). 5, 8, 13, 20, 29..
0.5, 2, 4.5, 8, 12.5...
7, 14, 25, 40, 59...
0, 13, 32, 57, 88...
4, -10, -32, -62, -100...
3, 12, 25, 42, 63...
-4, -9, -20, -37, -60...
2). 5). 8). 11). 14). 17). 20). 0, 3, 8, 15, 24...
2, 6, 12, 20, 30...
0, 7, 16, 27, 40...
11, 11, 9, 5, -1...
1.5, 5, 9.5, 15, 21.5...
-3.5, -1.5, 6.5, 20.5, 40.5..
6, 5, 3, 0, -4...
3). 6). 9). 12). 15). 18). 21). 2, 8, 18, 32, 50...
4, 10, 18, 28, 40...
-6, -6, -4, 0, 6...
3, 12, 31, 60, 99...
-4, 18, 54, 104, 168...
5.5, 11, 20.5, 34, 51.5...
2.25, 6.5, 12.25, 19.5, 28.25..
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