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Questions and Answers
For odd values of n, how is $(y_n)0$ expressed in terms of $(y{n-2})_0$?
For odd values of n, how is $(y_n)0$ expressed in terms of $(y{n-2})_0$?
- $(y_n)_0 = [(n-2)^2 + a^2](y_{n-2})_0$
- $(y_n)_0 = [(n+2)^2 - a^2](y_{n-2})_0$
- $(y_n)_0 = [n^2 - a^2](y_{n-2})_0$
- $(y_n)_0 = [(n-2)^2 - a^2](y_{n-2})_0$ (correct)
When n is even, what is the general solution for $(y_n)_0$?
When n is even, what is the general solution for $(y_n)_0$?
- $(y_n)_0 = 0$ (correct)
- $(y_n)_0 = a^2$
- $(y_n)_0 = n^2 - a^2$
- $(y_n)_0 = 1$
Given that $(y_3)_0 = (3^2 - a^2)(y_1)_0$ and $(y_1)_0 = a$, what is the expression for $(y_5)_0$?
Given that $(y_3)_0 = (3^2 - a^2)(y_1)_0$ and $(y_1)_0 = a$, what is the expression for $(y_5)_0$?
- $(y_5)_0 = (5^2 - a^2)a$
- $(y_5)_0 = (3^2 - a^2)(1 - a^2)a$
- $(y_5)_0 = (5^2 + a^2)(3^2 + a^2)a$
- $(y_5)_0 = (5^2 - a^2)(3^2 - a^2)a$ (correct)
For odd values of n, the expression for $(y_n)_0$ is a product of terms. What is the last term in this product?
For odd values of n, the expression for $(y_n)_0$ is a product of terms. What is the last term in this product?
If n is odd and $(y_1)_0 = a$, what does $(y_3)_0$ equal?
If n is odd and $(y_1)_0 = a$, what does $(y_3)_0$ equal?
When n is even, how does $(y_n)0$ relate to $(y{n-2})_0$?
When n is even, how does $(y_n)0$ relate to $(y{n-2})_0$?
If n is odd, which of the following represents the general form of $(y_n)_0$?
If n is odd, which of the following represents the general form of $(y_n)_0$?
For even values of n, what is a key characteristic of the solution for $(y_n)_0$?
For even values of n, what is a key characteristic of the solution for $(y_n)_0$?
When n is odd, what is the general pattern of terms in the expression for $(y_n)_0$?
When n is odd, what is the general pattern of terms in the expression for $(y_n)_0$?
What is the value of $(y_2)_0$ when n is even?
What is the value of $(y_2)_0$ when n is even?
If n is even, how does the value of a influence the general solution for $(y_n)_0$?
If n is even, how does the value of a influence the general solution for $(y_n)_0$?
If $(y_1)_0 = a$ and n is odd, which expression represents how the terms change recursively?
If $(y_1)_0 = a$ and n is odd, which expression represents how the terms change recursively?
Based on the pattern, what can we infer about the behavior of solving for even n values?
Based on the pattern, what can we infer about the behavior of solving for even n values?
If we know $(y_5)_0$ for a specific value of 'a' when n is odd, how can we determine $(y_7)_0$?
If we know $(y_5)_0$ for a specific value of 'a' when n is odd, how can we determine $(y_7)_0$?
What is a critical difference in the solutions for $(y_n)_0$ between even and odd values of n?
What is a critical difference in the solutions for $(y_n)_0$ between even and odd values of n?
When finding $(y_n)_0$ for odd values of n, how does increasing n affect the complexity of the expression?
When finding $(y_n)_0$ for odd values of n, how does increasing n affect the complexity of the expression?
For even n, what conclusion can be made about the value of $(y_n)_0$ regardless of the parameter a?
For even n, what conclusion can be made about the value of $(y_n)_0$ regardless of the parameter a?
What is the significance of the recursive relationship $(y_n)_0 = (n-2)^2 - a^2_0$ for odd n?
What is the significance of the recursive relationship $(y_n)_0 = (n-2)^2 - a^2_0$ for odd n?
What is the primary factor that determines the value of $(y_n)_0$ when n is even?
What is the primary factor that determines the value of $(y_n)_0$ when n is even?
Flashcards
(y_n)_0 when n is odd
(y_n)_0 when n is odd
When n is odd, the general form of (y_n)_0 involves the product of terms (k^2 - a^2) up to a final constant 'a'.
Recursive Formula for (y_n)_0
Recursive Formula for (y_n)_0
A recursive formula expressing (y_n)0 in terms of (y(n-2))_0, where the initial condition affects the subsequent values.
(y_n)_0 when n is even
(y_n)_0 when n is even
When n is an even number, (y_n)_0 equals 0.
What is (y_n)_0?
What is (y_n)_0?
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Study Notes
- Case 1: When n is odd
- n = 1, 3, 5, ..., (n - 2) in (4)
- When n = 1:
- (y3)0 = (1² - a²)(y1)0 = (1² - a²)a
- When n = 3:
- (y5)0 = (3² - a²)(y3)0 = (3² - a²)(1² - a²)a
- When n = 5:
- (y7)0 = (5² - a²)(y5)0
- (5² - a²)(3² - a²)(1² - a²)a
- Similarly, when n = n - 2:
- (yn)0 = (n - 2)² - a²0
- (yn)0 = [(n - 2)² - a²](n - 4)² - a²0
- (yn)0 = [(n - 2)² - a²][(n - 4)² - a²]...(3² - a²)(1² - a²)a
Case 2: When n is even
- n = 2, 4, 6, ..., (n - 2) in (4) we get
- When n = 2:
- (y4)0 = (2² - a²)(y2)0 = (2² - a²)*0 = 0, (y2)0 = 2
- When n = 4:
- (y6)0 = (4² - a²)(y4)0 = (4² - a²)*0 = 0
- When n = 6:
- (y8)0 = (6² - a²)(y6)0 = (6² - a²)*0 = 0
- Similarly, for all even values of n:
- (yn)0 = 0
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