Problem 1: Fibonacci Sequence

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1.

This is an interesting problem in the field of dynamic programming because the Fibonacci sequence can be efficiently calculated using dynamic programming techniques.

Let's take a look at how we can solve the Fibonacci sequence problem using dynamic programming. Here's the Java code that calculates the Fibonacci number at a given position n using dynamic programming:

1public class Fibonacci {
2
3    public static int fibonacci(int n) {
4        // base cases
5        if (n == 0) {
6            return 0;
7        }
8        if (n == 1) {
9            return 1;
10        }
11        // array to store fibonacci values
12        int[] fib = new int[n + 1];
13        fib[0] = 0;
14        fib[1] = 1;
15        // compute fibonacci sequence
16        for (int i = 2; i <= n; i++) {
17            fib[i] = fib[i - 1] + fib[i - 2];
18        }
19        // return the fibonacci number at position n
20        return fib[n];
21    }
22
23    public static void main(String[] args) {
24        int n = 10;
25        int fibValue = fibonacci(n);
26        System.out.println("The fibonacci value at position " + n + " is: " + fibValue);
27    }
28}

In this code, we define a fibonacci method that takes an integer n as input and returns the Fibonacci number at that position using dynamic programming.

The method first handles the base cases where n is 0 or 1, and then creates an array fib to store the Fibonacci values. We initialize the first two values of the array with 0 and 1, and then use a loop to compute the remaining Fibonacci values. Finally, we return the Fibonacci value at position n.

To test the code, we can call the fibonacci method with a value of n and print the result. For example, in the main method, we set n to 10 and print the Fibonacci value at position 10.

Feel free to modify the code and test it with different values of n to see the Fibonacci sequence in action!

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public class Fibonacci {

​

    public static int fibonacci(int n) {

        // base cases

        if (n == 0) {

            return 0;

        }

        if (n == 1) {

            return 1;

        }

        // array to store fibonacci values

        int[] fib = new int[n + 1];

        fib[0] = 0;

        fib[1] = 1;

        // compute fibonacci sequence

        for (int i = 2; i <= n; i++) {

            fib[i] = fib[i - 1] + fib[i - 2];

        }

        // return the fibonacci number at position n

        return fib[n];

    }

​

    public static void main(String[] args) {

        int n = 10;

        int fibValue = fibonacci(n);

        System.out.println("The fibonacci value at position " + n + " is: " + fibValue);

    }

}