Math Help - recurison + induction

1. recurison + induction

-1 for n greater/equal to 1

2. Prove by induction:

$f_{2}+f_{4}+.........+f_{2n}=f_{2n+1}-1$

Prove for n=1:

$f_{2(1)+1}-1=f_{3}-1=2-1=1$......true.

Assume $P_{k}$ is true:

$f_{2}+f_{4}+...........+f_{2k}=f_{2k+1}-1$

now, we must show that $P_{k+1}$ is true:

$f_{2}+f_{4}+...........+f_{2(k+1)}=f_{2(k+1)+1}-1$ is true.

$f_{2}+f_{4}+...........+f_{2k}+f_{2k+2}=\underbrac e{f_{2k+1}+f_{2k+2}}_{\text{this is f(2k+3)}}-1$

$f_{2k+1}+f_{2k+2}=f_{2k+3}$ and the proof is complete.