Algebra (Equation finding)

Question

I need help finding the equation. The points are (1,4) (2,10) (3,18) (4,28) (5,40) (6,54) (7,70) and (8,88). Thanks in advance :).

Answer 1

Take the differences between successive ordinates: 6, 8, 10, 12, 14, 18, 20.

The formula for these differences is 6 + 2(n-1) beginning with n = 1.

Getting from here to the answer is harder to explain; let me start with the answer: n² + 3n.

Before going any further you should check to make sure that you agree that the answer I've given is correct.

 

Now, why?

When you see that the differences between successive numbers in a sequence are constant, you know that the sequence is arithmetic.  When the difference form an arithmetic sequence you know that the sequence will be a quadratic function...and so on.

 

There is probably an easy way to explain this, but I always use the calculus of finite differences.

The first difference of n² = (n + 1)² - n² = 2n + 1 and the first difference of n = 1.

Consequently, 6n differences to 6 and and n² - 3n differences to 2n - 2 so n² + 3n differences to 6 + 2(n - 1)...which is what is needed for this problem.

This is a very short course of finite differences...and if there is a better way to explain the solution, I do not know it!