Find the sum of the first n odd numbers, 1+ 3 + … + 2n-1, in terms of n. What might a mathematician’s thought process be?

[Answer for a Further Maths A level student]: We could just apply the formula for summing an arithmetic series, but this won’t give a good understanding of why the answer we get is correct, which is always important in maths. So let’s do it from scratch: by calculating the sum in question for some small values of n (when unsure, mathematicians usually experiment), we quickly conjecture (guess) that the answer is n². But no matter how many values of n we check, a mathematician will not be satisfied until we have a proof for all n. One way to do this is by induction. In this case, this is simple: Base case: the result is certainly true for n=1. Assume that 1+ 3 + … + 2k-1 = k² Induction: Then 1+ 3 + … + 2k-1 + 2k+1 = k² + 2k + 1 = (k+1)² Conclusion: Result is true for all n. But why did the factorisation happen so nicely in the induction step? Draw a k by k square grid. We can see that to turn this into an (k+1) by (k+1) grid, we need to add 2k+1 little squares. This idea gives a lovely way to divide an n by n grid as 1+ 3 + … + 2n-1, allowing us to “see” the answer to the problem. A mathematician might use this insight to discover more things, like the harder result that every power n^p of a number n is the sum of consecutive odd numbers (hint: consider an n by n^p-1 grid).