For each of the following sequences, find an explicit and a recursive description of the pattern. Assume sequences start from n = 1.
a. 1, -1, -3, -5... b. 8,2,3,9...
c. -2, 0, 0, -2, -6...
a) note that you just keep subtracting 2, so
an = 1+2(n-1) = 2n-1
b) Hmmm. Differences are
8, 2, 3, 9, ...
.-6, 1, 6, ...
....7, 5 ...
Not a quadratic. The terms are
8, 8(1/4), 2(6/4), 3(12/4), ...
The numerators are 1,6,12. Again, not much good there.
The sequence is very short, so there are lots of rules that can work on just these four numbers. Note that if we subtract 1 from the last two terms, they are the same as the first two terms. So, we might say
a1 = 8
a2 = 2
an+2 = 1+a5-n
Not very satisfying. Maybe you can pierce the gloom.
(c) Note that the values are symmetric about an axis between the 0's. That indicates a quadratic, with an axis of symmetry at x = 5/2. So, we get
y = a(x - 5/2)^2 + k
Since y(2)= 0 and y(1) = -2,
a/4 + k = 0
9a/4 + k = -2
a = -1, k=1/4
y = -(x - 5/2)^2 + 1/4
= -(x-2)(x-3)
or,
an = -(n-2)(n-3)
a. To find an explicit description of the pattern in sequence a, we need to observe the relationship between the terms.
Looking at the sequence, we notice that each term is decreasing by 2 from the previous term. We can express this relationship as:
Term(n) = 1 - (n-1) * 2
In this formula, "n" represents the position of the term in the sequence. For example, the first term is at position n = 1, the second term at n = 2, and so on.
For the recursive description, we can define the base case as follows:
Term(1) = 1
And the recursive formula can be defined as:
Term(n) = Term(n-1) - 2
This means each term is obtained by subtracting 2 from the previous term in the sequence.
b. For sequence b, let's analyze the pattern to find an explicit and recursive description.
Looking at the sequence, we see no obvious constant difference between the terms. However, if we look closer, we notice that the difference between the first and second terms (8 - 2) is 6, and the difference between the second and third terms (2 - 3) is -1. This suggests that there may be an alternating pattern at play.
Based on this observation, we can describe the pattern explicitly as follows:
Term(n) = 8 + (-1)^(n-1) * (n-1)
In this formula, "^" represents exponentiation, and "^(n-1)" is used to alternate the sign based on whether n is odd or even.
For the recursive description, we can define the base case as:
Term(1) = 8
And the recursive formula can be defined as:
Term(n) = Term(n-1) + (-1)^(n-1) * (n-1)
This means each term is obtained by adding or subtracting the value of (n-1) to the previous term, depending on whether n is odd or even.
c. Let's analyze the pattern in sequence c to find the explicit and recursive descriptions.
Looking at the sequence, we notice that the terms alternate between -2 and 0. Additionally, the terms also increase in magnitude by 2 each time. We can describe this pattern explicitly as follows:
Term(n) = (-1)^(n+1) * 2 - 2
In this formula, "^" represents exponentiation, and "^(n+1)" is used to alternate the sign based on whether n is odd or even.
For the recursive description, we can define the base cases as:
Term(1) = -2
Term(2) = 0
And the recursive formula can be defined as:
Term(n) = Term(n-2) - 2
This means each term is obtained by subtracting 2 from the term that is two positions behind in the sequence.