Let ƒÆ=sin −1 7 25 . Consider the sequence of values defined by a n =sin(nƒÆ) . They satisfy the recurrence relation
a n+2 =k 1 a n+1 +k 0 a n ,n�¸N
for some (fixed) real numbers k 1 ,k 0 . The sum k 1 +k 0 can be written as p q , where p and q are positive coprime integers. What is the value of p+q ?
To find the value of p+q, we need to determine the values of k1 and k0.
Given the recurrence relation an+2 = k1an+1 + k0an, we can rewrite it as a characteristic equation by assuming an = λ^n. Substituting this into the recurrence relation, we get:
λ^(n+2) = k1λ^(n+1) + k0λ^n
Dividing both sides by λ^n, we get:
λ^2 = k1λ + k0
This is a quadratic equation in λ. To solve this equation, we can use the Quadratic Formula:
λ = (-k1 ± √(k1^2 - 4k0))/2
Since λ represents the roots of the characteristic equation, it determines the behavior of the sequence. The values of λ can be real, complex, or repeated.
Now, let's calculate the value of λ. Given ƒÆ = sin^(-1)(7/25), we can find its value using a calculator:
Į = sin^(-1)(7/25) Š0.2818 (in radians)
Next, we substitute this value into the characteristic equation:
λ^2 = k1λ + k0
(0.2818)^2 = k1 * 0.2818 + k0
0.0794 = 0.2818k1 + k0
This equation gives us a relationship between k1 and k0. By solving this equation, we can find the values of k1 and k0.
To summarize, to find the values of k1 and k0, we need to solve the equation (0.2818)^2 = k1 * 0.2818 + k0. Once we determine the values of k1 and k0, we can calculate k1 + k0 and then find p + q by writing it in the form of p/q, where p and q are positive coprime integers.