Which sequence has constant 2nd differences?
A sequence that has constant 2nd differences is called a quadratic sequence. In a quadratic sequence, the differences between consecutive terms are not constant, but the differences between those differences are constant.
An example of a quadratic sequence is:
1, 4, 9, 16, 25, ...
The differences between consecutive terms are: 3, 5, 7, 9, ...
The differences between those differences are all constant: 2, 2, 2, ...
A sequence has constant 2nd differences if the differences between consecutive terms are constant. To find a sequence with constant 2nd differences, we need to find an equation that represents the pattern.
Let's start by finding the differences between consecutive terms for a few sequence examples and see if any of them have constant differences:
Example 1: 1, 3, 5, 7, 9
Differences between consecutive terms: 3-1=2, 5-3=2, 7-5=2, 9-7=2
The differences between consecutive terms in this sequence are constant (2), so this sequence has constant 1st differences. However, we need to check if the 2nd differences are also constant.
Differences between the differences: 2-2=0, 2-2=0, 2-2=0
The differences between the differences in this sequence are constant (0). Therefore, this sequence has constant 2nd differences.
Example 2: 2, 5, 10, 17, 26
Differences between consecutive terms: 5-2=3, 10-5=5, 17-10=7, 26-17=9
The differences between consecutive terms in this sequence are not constant, so this sequence does not have constant 1st differences. Therefore, we can conclude that it does not have constant 2nd differences either.
Example 3: 1, 4, 9, 16, 25
Differences between consecutive terms: 4-1=3, 9-4=5, 16-9=7, 25-16=9
The differences between consecutive terms in this sequence are not constant, so this sequence does not have constant 1st differences. Therefore, we can conclude that it does not have constant 2nd differences either.
From the examples above, we can see that the sequence 1, 3, 5, 7, 9 has constant 2nd differences.
To determine which sequence has constant 2nd differences, we need to first understand what 2nd differences are.
The 2nd differences of a sequence are obtained by taking the differences between consecutive terms in the differences sequence. In other words, it is the difference between the differences themselves.
Let's consider an example sequence: 1, 4, 9, 16, 25, 36.
To find the differences, we subtract each term from its consecutive one:
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11
Now, we find the differences between these differences:
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
11 - 9 = 2
Since the differences between the differences are all the same (2) in this sequence, it has constant 2nd differences.
However, let's consider another sequence: 2, 5, 10, 17, 26, 37.
Finding the differences:
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
26 - 17 = 9
37 - 26 = 11
Then, finding the differences between these differences:
5 - 3 = 2
7 - 5 = 2
9 - 7 = 2
11 - 9 = 2
In this sequence, the differences between the differences are also constant (2). Hence, this sequence has constant 2nd differences.
Therefore, both the example sequence 1, 4, 9, 16, 25, 36 and the sequence 2, 5, 10, 17, 26, 37 have constant 2nd differences.