The Lucas sequence $L_1,L_2,$dots $($named after $$douard Lucas,

the inventor of the Tower of Hanoi puzzle$)$ is defined by the

recurrence relation

$L_n=L_{n-1}+L_{n-2},n3,$

and the initial conditions $L_1=1,L_2=3.$

$($a$)$ Find the values of $L_3,L_4,$ and $L_5.$

$($b$)$ Show that

$L_{n+2}=f_{n+1}+f_{n+3},n1,$

where $f_1,f_2,$dots denotes the Fibonacci sequence.