Write a program to find the nth Fibonacci number using dynamic programming.

Great ! I ' m happy to help you with that . \n To find the n th F ib on acci number using dynamic programming , we can use the following approach : \n 1 . Initialize two arrays , ` f ib on acci ` and ` f ib on acci _ cache `, with sizes equal to the number of elements we want to compute the F ib on acci sequence up to . \n 2 . Initialize the values of ` f ib on acci ` and ` f ib on acci _ cache ` to 0 and 1 , respectively . \n 3 . For each element in the sequence , use the following rec urrence relation : ` f ib on acci [ i ] = fib on acci [ i - 1 ] + fib on acci [ i - 2 ]`. \n 4 . Use the ` f ib on acci _ cache ` array to store the previously computed values of the F ib on acci sequence , so that we don ' t have to re compute them unnecess arily . \n 5 . When we reach the end of the sequence , return the value of ` f ib on acci [ n ]`. \n Here is some sample Python code that implements ates this approach : \n ``` \n def fib on acci ( n ): \n # Initialize the fib on acci and fib on acci _ cache arrays \n fib on acci = [ 0 , 1 ] * ( n + 1 ) \n fib on acci _ cache = [ 0 ] * ( n + 1 ) \n \n # Rec urs ively compute the F ib on acci sequence \n for i in range ( 2 , n + 1 ): \n fib on acci [ i ] = fib on acci [ i - 1 ] + fib on acci [ i - 2 ] \n \n # Return the value of fib on acci [ n ] \n return fib on acci [ n ] \n \n # Example usage : \n print ( f ib on acci ( 5 )) # Output : 5 \n print ( f ib on acci ( 8 )) # Output : 2 1 \n ``` \n I hope this helps ! Let me know if you have any questions or need further clar ification . </s>