Core pure maths Series question

Images and question below

My question is about part b)


Because r=11 instead of r=1, we subtract the r^3 expression where n = 30 and r^3 expression where n = 10. This is my working out



However…this isn’t the correct solution. The correct solution is:

This solution doesn’t find 30 from 11, it merely finds a solution when n=10. I’m thinking that perhaps the solution is incorrect because it doesn’t make sense at all, but I don’t want to get ahead of myself

Once youve shown the first part, you simply use it with n=10 in the second part as that corresponds to a lower limit of 11 and upper limit of 30.

Why here? In previous examples, the standard solution has always been to substract both expressions. What’s special about this case that the solution goes against the standard rule?

Why here? In previous examples, the standard solution has always been to substract both expressions. What’s special about this case that the solution goes against the standard rule?

(side-sidenote, it's actually the same as the square of the sum from 1 to n

In the first part youve shown that the cubic sum which starts at (n+1)^3 and finishes at (3n)^3 is the given quartic in n. So this question part has done the operation of evaulating the cubic sum from 1 to 3n then subtracting the cubic sum of 1 to n and then simplify, which you must have done.

The second part is a "hence" so youre expected to use the first part starting at 11^3 and finishing at 30^3 which corresponds to n=10. If you used the standard sum of cubes starting at 1, youd simply repeat the work you did in part a). Note that sometimes its worth writing out a few of the terms in the summation at the start (lower limit) and end (upper limit) to make sure youre interpreting it properly.

Thank you for your reply! I got confused here, what is the same as the sum of 1 to n? Bear in mind I haven’t covered proof by induction yet so perhaps that’s why I’m confused