Core Pure AS Maths: series question

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Here is the question and here is the solution for the question. I don’t understand the solution

So n + k is set to equal to m, and then m is put into the r^3 formula, which is then set equal to n + k. Why not put n + k into the r^3 formula directly? Setting n+k equal to m and then putting m into the formula only to get rid of the m, all seems unnecessary

The solution is pretty much as you describe, so use the usual sum of cubes expression where you replace the upper limit m with n+k. The first and second lines should probably be swapped and the let m=n+k is pretty much unnecessary.

I see. So the most efficient way to answer the question is to just substitute n+k into the cubic expression and nothing else. Thank you!

Yes and no. I would say that a model answer here (*) should state (in some form or other) the result you're assuming [i.e. $\sum_{k=1}^n k^3 = \frac{1}{4}n^2(n+1)^2$], and it should end with the result you're proving [i.e. $\sum_{r=1}^{n+k} r^3 = \frac{1}{4}(n+k)^2(n+k+1)^2$].

Personally, I think you need at least a smidgen of explanation as well, otherwise you've literally quoted a standard result and written the answer. So you'd need to say "substituting n+k for n" or similar (and note that this is slightly awkward language, because really the "n" in "n+k" is something different from the "n" in "for n", so you're using the same letter for 2 things).

In practical terms, the given solution is very direct - by quoting the "sum of cubes" result using m instead of n it means you can just say "set m = k+n" with no other explanation required.

(*) In general, answers to maths questions should still be "full sentences", in the same way as for more essay based subjects. (Time constraints may make you take shortcuts, but a model answer shouldn't). In this case that means there's a minimal length it's hard to reasonably go below.