$7.$ Consider the Lucas numbers:

Ln $=$ Ln$1+$ Ln$2,$ n $>1$; L$1=1$; L$0=2.$

a$)$ Write a recursive function to compute Ln using the above definition directly. Implement

your solution and print Li$5,$ where $0<=$ i $<=10,$ as output.

b$)$ Write a recursive function$/$procedure to compute Ln with time complexity O$($n$)($more

precisely, the time complexity should be O$($nA$($n$))$ when n is large, where A$($n$)$ is the

complexity of adding Ln$1$ and Ln$2).$ Implement your solution and print Li$20,$ where

$0<=$ i $<=25,$ as output. This program must be able to compute Ln precisely for n $<=500.$

Hint $1$:

Let Kn $=$

Ln

Ln$1$

: Kn $=$

$11$

$10$

$\backslash$times Kn$1,$ n $>=1$; K$0=$

$2$

$1$

With this formulation, design a recursive algorithm for Kn such that the algorithm will

return both Ln and Ln$1$ with input parameter n$.$

Hint $2$: Can you use a primitive type to store L$500?$

For programs in $7$ a$)$ and $7$ b$)$ of this question, you are NOT allowed to use Python. For

C$++$ and Java, you can only use primitive types such as char, int, long and long long.

You are not allowed to use large integer, such as BigInteger in Java, from the language

library. You have to write your own large integer class or data type, if needed.