How do you find the rule of x and y?

Question

x:1,2,3,4,5,13,15

y:-1,2,7,14,23

Answer 1

If a sequence {a_n} is generated by a polynomial of degree n, then the differences of consecutive terms {b_n = a_n+1 - a_n} (called the first differences) form a sequence generated by a polynomial of degree n-1. The n-th differences sequence is constant

 

-1   2    7    14    23  ← Your sequence

    3   5    7     9       ← First differences

      2    2    2           ← Second differences

 

So your function is quadratic f(x) = ax² + bx + c

 

f(1)=-1, f(2)=2, and f(3) = 7 gives a system

 

(1)    a +b +c = -1

(2)  4a+2b+c = 2

(3)  9a+3b+c = 7

 

Let's solve the system.

 

(4)=(2)-(1)     3a + b = 3

(5)=(3)-(2)     5a + b = 5

(6)=(5)-(4)     2a = 2

 

a = 1

b = 3 - 3a = 3-3·1 = 0

c = -1 - a - b = -1 - 1 - 0 = -2

 

So y = x² - 2

 

When x=13, y=13²-2 = 167.

When x=15, y=15²-2 = 223.