Help, I can't find the pattern rule for this sequence(eg. -6,-3,2,9,18 and the pattern rule is n squared-7)

The pattern: 6,28,64,114,178.
What's the pattern rule?

The pattern: 96,84,64,36.
What's the pattern rule?

The pattern: 0,24,216,960.
What's the pattern rule?

These appear to be susceptible to solution using Newton's Divided Difference method.

For the first one,

n: 1 2 3 4 5
f(n): 6 28 64 114 178
d1(n)=f(n+1)-f(n): 22 36 50 64
d2(n)=[d1(n+1)-d1(n)]/2: 7 7 7 7

The second differences are constant. So, the final function has coefficients from the first entry in each row:

f(n) = 6 + 22(n-1) + 7(n-1)(n-2)
f(n) = 7x^2 + x - 2

This is the polynomial of lowest degree which fits all the given data points.

Read up on the method, and you can do the others easily. A web search on "divided difference interpolation formula" reveals lots of articles, and even a calculator at the ucla site.

pattern rule for

2,3, ___, 7, 11, 13, 17, ___

8,9,11,14 , , ,

15,10,5 I need A. B. C. I need the number pattern

BOOTY

BOOTYHOLE

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To find the pattern rule for a sequence, you need to analyze the differences between consecutive terms and then determine the relationship between those differences. Let's work through each sequence one by one.

Sequence 1: -6, -3, 2, 9, 18.
To find the pattern rule for this sequence, let's first calculate the differences between consecutive terms:
-6 - (-3) = -3
-3 - 2 = -5
2 - 9 = -7
9 - 18 = -9

We can observe that the differences are decreasing by 2 units each time. Now, let's analyze the relationship between the terms and the differences.
The differences are decreasing by 2 units each time, implying a quadratic relationship. So, the pattern rule for this sequence can be expressed as n^2 - 7, where n is the position of the term in the sequence.
Thus, the pattern rule for this sequence is n^2 - 7.

Sequence 2: 6, 28, 64, 114, 178.
Now, let's calculate the differences between consecutive terms:
28 - 6 = 22
64 - 28 = 36
114 - 64 = 50
178 - 114 = 64

We can see that the differences are increasing by 14 units each time. Now, let's analyze the relationship between the terms and the differences.
The differences are increasing by 14 units each time, indicating a linear relationship. So, the pattern rule for this sequence can be expressed as 14n + 6, where n is the position of the term in the sequence.
Thus, the pattern rule for this sequence is 14n + 6.

Sequence 3: 96, 84, 64, 36.
Let's calculate the differences between consecutive terms:
84 - 96 = -12
64 - 84 = -20
36 - 64 = -28

The differences are decreasing by 8 units each time. Analyzing the relationship between the terms and the differences, we can observe a combination of linear and quadratic relationships. Thus, we need to formulate a pattern rule that incorporates both.
One way to express this pattern rule is as follows: n^3 - 20n^2 + 118n - 18, where n is the position of the term in the sequence.
Thus, the pattern rule for this sequence is n^3 - 20n^2 + 118n - 18.