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If a function F is such that F(0)=2, F(1...

If a function `F` is such that `F(0)=2`, `F(1)=3`, `F(n+2)=2F(n)-F(n+1)` for `n ge 0`, then `F(5)` is equal to

A

`-7`

B

`-3`

C

`7`

D

`13`

Text Solution

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The correct Answer is:
To find \( F(5) \) given the function defined by \( F(0) = 2 \), \( F(1) = 3 \), and the recurrence relation \( F(n+2) = 2F(n) - F(n+1) \) for \( n \geq 0 \), we will calculate the values step by step. ### Step 1: Calculate \( F(2) \) Using the recurrence relation: \[ F(2) = 2F(0) - F(1) \] Substituting the known values: \[ F(2) = 2 \cdot 2 - 3 = 4 - 3 = 1 \] ### Step 2: Calculate \( F(3) \) Now we can find \( F(3) \): \[ F(3) = 2F(1) - F(2) \] Substituting the known values: \[ F(3) = 2 \cdot 3 - 1 = 6 - 1 = 5 \] ### Step 3: Calculate \( F(4) \) Next, we find \( F(4) \): \[ F(4) = 2F(2) - F(3) \] Substituting the known values: \[ F(4) = 2 \cdot 1 - 5 = 2 - 5 = -3 \] ### Step 4: Calculate \( F(5) \) Finally, we calculate \( F(5) \): \[ F(5) = 2F(3) - F(4) \] Substituting the known values: \[ F(5) = 2 \cdot 5 - (-3) = 10 + 3 = 13 \] Thus, the value of \( F(5) \) is \( \boxed{13} \).
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