--THINK AHEAD--
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"Because even though they had nothing in common on the surface, their second differences brought them together."
--DEFINITION--
"What is a quadratic sequence?"
The nth term of a quadratic sequence comes in the form of an^2 + b, which means that, unlike the linear sequence, the differences in the sequence are not the same (not linear), HOWEVER, the difference of the differences (second difference) are often constant."
For example in the sequence 3, 6, 11, 18, 27 …
The first differences are 3, 5, 7, 9, so they are nonlinear, but...
The second differences are 2, 2, and 2 which are constant
"Now let's work out the nth term for a quadratic step-by-step!"
--WORKED EXAMPLE--
Let's use the sequence above: 3, 6, 11, 18, 27
Now let's put it in the form an^2 + b
Step 1: Identify the first difference
For 3, 6, 11, 18, 27 the first difference are
3, 5, 7, 9,
Step 2: Identify the second difference
For 3, 5, 7, 9 the 2nd difference are
2, 2, and 2
Step 3: Work out the value of 'a' using step 2
a = second difference / 2
We worked out that the second difference = '2'
So in this sequence a = '2'/2 = 1
So, a = 1
Step 4: Work out the value of 'b'
b = difference between actual sequence and the value of the an^2 sequence.
Now that we know that the an^2 sequence is (1)n^2, its sequence is 1, 4, 9, 16, 25, so the difference between the actual sequence and these values is the value of 'b'.
Actual: 3, 6, 11, 18, 27
n^2: 1, 4, 9, 16, 25
Diff (='b'): = 2
So, b = 2
Step 5: Work out the nth term
We have:
n^2 + 2
"Complete these worksheets using the guidance below."
--WORKSHEETS--
Sometimes the 'sequence minus n^2' (Step 5) is NOT the same number; it is, in fact, another 'linear sequence' that needs solving to get the 'linear nth term.' See the example below.
"Choose one of your completed sequences from the worksheets and show your process using the exact 6-step method you can see in my example below."