The main culprit is that you have put this
int sum=0;
inside the for loop, the effect of which is this sum is being reset to zero on each iteration, that results in having to repeat the previous day's addition in subsequent iteration again and again. So place it before the start of the loop. Once you do that, the sum will retain its value from the preceding iteration, that will get rid of the redundant addition in each if block.
Next, each of the if condition is based on the index number of the array, which you can derive from the iteration count which is y in your code, so why re-invent the wheel. You only need one line of code like this:
sum = sum + costsarr[y] * (y+1);
as the iteration count will take care of the array index.
Putting together, the final code should look like:
int[] costsarr = {210, 375, 181, 600, 750, 360, 13125, 58, 7553, 5509, 2708, 2934};
int sum = 0;
for(int i=0;i<costsarr.length;i++)
{
int day = i+1;
sum += costsarr[i] * day;
System.out.println("Day " + day + ": $" + sum);
}
Enjoy your learning...