Answer :
From the given problem :
f(1) = 2
f(n) = -2f(n-1) + 3
f(5) = ?
Let's find f(2) up to f(5).
For f(2), n = 2
[tex]\begin{gathered} f(n)=-2f(n-1)+3 \\ f(2)=-2f(1)+3 \\ f(2)=-2(2)+3 \\ f(2)=-4+3 \\ f(2)=-1 \end{gathered}[/tex]For f(3), n = 3
[tex]\begin{gathered} f(n)=-2f(n-1)+3 \\ f(3)=-2f(3-1)+3 \\ f(3)=-2f(2)+3 \\ f(3)=-2(-1)+3 \\ f(3)=2+3 \\ f(3)=5 \end{gathered}[/tex]For f(4), n = 4
[tex]\begin{gathered} f(n)=-2f(n-1)+3 \\ f(4)=-2f(4-1)+3 \\ f(4)=-2f(3)+3 \\ f(4)=-2(5)+3 \\ f(4)=-10+3 \\ f(4)=-7 \end{gathered}[/tex]For f(5), n = 5
[tex]\begin{gathered} f(n)=-2f(n-1)+3 \\ f(5)=-2f(5-1)+3 \\ f(5)=-2f(4)+3 \\ f(5)=-2(-7)+3 \\ f(5)=14+3 \\ f(5)=17 \end{gathered}[/tex]The answer is f(5) = 17
