if we have pentagonal numbers 1 5 12 22 35 fit a quadratic model to find a formula for p(n) the nth pentagonal number.
To fit a quadratic model and find a formula for the nth pentagonal number, we can start by observing the pattern between the given pentagonal numbers:
1, 5, 12, 22, 35
We recognize that the difference between consecutive pentagonal numbers increases by 3 each time. This indicates that the formula for the nth pentagonal number, p(n), involves a quadratic function with a coefficient of n^2.
Now, let's write down the general form of a quadratic equation:
p(n) = An^2 + Bn + C
To find the values of A, B, and C, we need to substitute the given pentagonal numbers into this equation. Let's use the first three numbers:
When n = 1, p(1) = 1
When n = 2, p(2) = 5
When n = 3, p(3) = 12
By substituting these values into the equation, we get a system of three equations:
Equation 1: p(1) = A(1)^2 + B(1) + C = 1
Equation 2: p(2) = A(2)^2 + B(2) + C = 5
Equation 3: p(3) = A(3)^2 + B(3) + C = 12
By solving this system, we can determine the values of A, B, and C.