Interview Questions for Minimum Cost of Tasks
Equivalence to another question
Here is how we can find the problem in the question is basically the same as the problem in another question. Hence the answer over there can be adapted easily to this problem.
     the problem here &lt;---&gt; the problem over there               
           home tasks &lt;---&gt; weekly shipments
cost c[i] by yourself &lt;---&gt; cost s_i * r by company A
         next 4 tasks &lt;---&gt; next 4 weeks
   cost h by handyman &lt;---&gt; cost 4c by company B
         minimum cost &lt;---&gt; minimum cost

For example, for cost array [1,3,5,6,7,2,1,3,1000] and handyman cost  11 here, we have the corresponding problem of $r=1$, frights by pounds[1,3,5,6,7,2,1,3,1000] and $c=11/4=3.75$.
You might want to ask why or how I can connect these two problems together. A simple reason is that I wrote the answer to the other question. Another reason is that I have been solving, modifying and creating problems about algorithms constantly. I would encourage you to try that kind of exercises from time to time.
Exercises
Exercise 1. Imagine a different backdrop of this problem.
 
Exercise 2. (A slight generalization) Let us have the same problem as in the question, the same list of $n$ consecutive tasks and a handyman with the following change. Besides 4 consecutive tasks of the fixed cost $h$, the handyman can also do the next 3 consecutive tasks on your list for the fixed cost of $g$, regardless of how much time and effort those 3 tasks would cost you. Find an $O(n)$-time recursive algorithm that calculates the lowest cost for completing the tasks. (This problem degenerates to the original problem if we let $g$ be sufficiently large.)
Here is the idea to solve exercise 2.
Let $dp[i]$ be the lowest cost to do tasks 1 to $i$ inclusive. The base cases and the recurrence relation for $dp[i]$ is given by the following description.

When $1 \leq i \leq 2$, you have to do all the tasks by yourself.
When $i=3$, either you do all the tasks or let the handyman does all 3 tasks.
When $i\geq 4$, the arrangement of lowest cost to complete all tasks must be one of the following ways.

You will do the last task by yourself. Complete the other $i-1$ tasks with the lowest cost.
Let the handyman do the last three tasks with cost $g$. Complete the other $i-3$ tasks with the lowest cost.
Let the handyman do the last four tasks with cost $h$. Complete the other $i-4$ tasks with the lowest cost.