The question is, **"Given an array A of n integers, you need to find the maximum sum of any contiguous subarray. For instance, the maximum sum of any contiguous subarray in the array -1, 2, 3, -2, 5, -6, 7, -8 is 9 (which is the sum of the subarray 2, 3, -2, 5, 6, 7). Complete the following Dynamic Programming solution for the above problem:**

**DP[0] = A[0]**

**For i = 1 to n**

** DP[i] = max(A[i],____)"**

This is the problem statement of Kadane's Algorithm.

To solve the problem of finding the maximum sum of any contiguous subarray using a dynamic programming approach, we can use the following logic:

**Initialization**:

Define an array

**DP**where**DP[i]**will store the maximum sum of any contiguous subarray that ends at index**i**.Initialize

**DP[0]**to**A[0]**because the maximum sum of a subarray ending at the first element is just the first element itself.

**Dynamic Programming Transition**:

For each element

**A[i]**(where**i**ranges from 1 to n-1), we need to decide whether to include it in the existing subarray (that ends at**i-1**) or start a new subarray from**A[i]**.Thus,

**DP[i]**should be the maximum of:**A[i]**itself (which means starting a new subarray from**A[i]**)**DP[i-1] + A[i]**(which means extending the subarray ending at**i-1**to include**A[i]**).

So, the dynamic programming relation can be written as:

𝐷𝑃[𝑖]=max(𝐴[𝑖],𝐷𝑃[𝑖−1]+𝐴[𝑖])

## Comments