Quadratic Sequences

1
Quadratic Sequences
2
Remember

• In linear Sequences, there is a constant
  difference between the terms
• This difference is said to be in the first
  layer.
• In Quadratic Sequences, the constant difference
  is in the second layer

3
Quadratic Sequences

• Example 1
• 6, 12, 20, 30, 42

• That makes this a quadratic sequence.
• We will try to find the rule for the nth term
  in the sequence by factoring
• Factor each number in the sequence looking for a
  pattern or common difference among the factors.

6
8
10
12
Layer 1
2
2
2
Layer 2
Notice that the difference between consecutive
terms in layer 1 is NOT the same.
But, the differences in layer 2 are the same.
4
6, 12, 20, 30, 42
2 x 3
3 x 4
4 x 5
5 x 6
6 x 7
Six can be factored as 1 x 6 or 2 x 3. 12 can
be factored many ways. But, if I factor it using
3 x 4, I start seeing a pattern among the
factors. Look
Look at the first of each set of factors.
At the same time, look at the second number of
each set of factors.
5
6, 12, 20, 30, 42
2 x 3
3 x 4
4 x 5
5 x 6
6 x 7
Look at the first number of each set of factors.
2 3 4 5 6
Use the information from linear sequences to find
the rule to Get this number They have a common
difference of one, so it would start with 1n (n
would represent the position of the term in the
sequence). Then, you would have to add one to
get each number. So the first number if each pair
is found by the rule n 1
6
6, 12, 20, 30, 42
2 x 3
3 x 4
4 x 5
5 x 6
6 x 7
Look at the second number in each pair of factors.
3 4 5 6 7
These numbers have a difference of one as well.
We start with 1n (Again, n is the position of
the number in the sequence). Then, you must add
2 to get the number. So this is n 2.
The rule for this whole sequence is
(n 1)
(n 2)
n2 3n 2
Or
7
Try another sequence

• 4, 15, 30, 49, 72,
• Factor the terms looking for a pattern in both
  numbers in each pair of factors.
• 1 x 4 3 x 5 5 x 6 7 x 7 9 x 8
• 1, 3, 5, 7, 9 Write the rule for the first
  number in each pair
• 2n - 1

8
4, 15, 30, 49, 72,

• 1 x 4 3 x 5 5 x 6 7 x 7 9 x 8
• 4, 5, 6, 7, 8 Write a rule for the second
  number in each pair of factors.
• Put the two rules together with multiplication.

9
Try these

• 6, 32, 78, 144,
• -5, 8, 29, 58, 95,
• -6, -4, 2, 12, 26, ...
• 4, 19, 44, 79, 124, ...