What would be the explicit rule if an equation started at 13 and added odd consecutive integers starting with 7?
+130514 So...I assume you mean the succession of terms.... is:
13 20 29 40 53 68 .....
We can use something called the sum of differences to find a polynomial that will model this
13 20 29 40 53 68
1st differences 7 9 11 13 15
2nd differences 2 2 2 2
We will have a 2nd degree polynomial with the following equations
a + b + c = 13 (1)
4a + 2b + c = 20 (2)
9a + 3b + c = 29 (3)
Subtracting (1) from (2) and (1) from(3) we end up with the following system
3a + b = 7 3a + b = 7 (4)
8a + 2b = 16 → 4a + b = 8 (5)
Subtracting (4) from (5) we have
a = 1 and using (4), we have, 3(1) + b = 7 so b = 4 and from (1), 1 + 4 + c = 13 so c = 8
And our polynomial that will generate our series is given by
P(n) = n^2 + 4n + 8
Notice that when n = 1, we get the first term, 13
When n= 2, we get the second term, 20
When n = 3 → 29
Etc.......
Here's a graph : https://www.desmos.com/calculator/qr5wottz54
