Question

If \(F\) is function such that \(F(0)=2,\ F(1)=3\), and
\[ F(x+2)=2F(x)-F(x+1)\ \text{for}\ x\geq 0, \] then \(F(5)\) is equal to

• A. \(-7\)
• B. \(-3\)
• C. \(17\)
• D. \(13\)

Hint:

For recurrence-based questions, compute values step-by-step using given base values. Always maintain correct substitution order.

Answer 1

The Correct Option is D

Step 1: Write the recurrence relation.
\[ F(x+2)=2F(x)-F(x+1) \] We are given:
\[ F(0)=2,\quad F(1)=3 \] Step 2: Compute \(F(2)\).
Put \(x=0\):
\[ F(2)=2F(0)-F(1)=2(2)-3=4-3=1 \] Step 3: Compute \(F(3)\).
Put \(x=1\):
\[ F(3)=2F(1)-F(2)=2(3)-1=6-1=5 \] Step 4: Compute \(F(4)\).
Put \(x=2\):
\[ F(4)=2F(2)-F(3)=2(1)-5=2-5=-3 \] Step 5: Compute \(F(5)\).
Put \(x=3\):
\[ F(5)=2F(3)-F(4)=2(5)-(-3)=10+3=13 \] Final Answer: \[ \boxed{13} \]