Question

I need help with this problem

Answer 1

Answer: 91, 140, 205

From 1 to 5 you add 2 squared
From 5 to 14 you add 3 squared
From 14 to 30 you add 4 squared
30 goes to 55
55 goes to 91
91 goes 140
140 goes to 205

Answer 2

91, 140, 204

Explanation:

To use the method of finite differences to find the next terms of the sequence, you must determine the differences between the terms, and the differences of those, down to the level where the differences are constant.

Then those values are propagated back upward to find the next terms of the sequence.

In the attachment, the numbers on each row are such that the row below is the difference of adjacent terms.

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Additional comment

Here, the 3rd row has differences that are constant. That means the original sequence can be defined by a 3rd-degree polynomial. That polynomial can be written ...

y = (x/6)(2x^2 +3x +1) . . . . where x is the term number: 1, 2, 3, ...