B.5:5.7 - Recover an argument, then change its starting point

A reader can use elementary algebra and wants to understand and adapt the claim that the sum of the first n positive odd integers is n squared, for a nonnegative integer n. Let S(n) denote that sum, with S(0) = 0.

Work backward from the formula. It is enough to establish its initial value and how it changes when one term is added. The next odd integer after the first n terms is 2n + 1, so S(n + 1) = S(n) + 2n + 1. The proposed value has the same change: (n + 1) squared - n squared = 2n + 1. Both start at zero. Repeating that step establishes S(n) = n squared for every finite n.

A square of n by n unit cells makes the same step visible. Add one row of n cells and an adjoining column of n + 1 cells; the resulting square has side n + 1. The added cells give 2n + 1. Counting the cells and the algebraic recurrence explain the same increase in different expressions.

Now the requested sum has n terms beginning at 3: 3 + 5 + … + (2n + 1). Recover which premise changed. These are the first n + 1 positive odd integers with the initial 1 removed. The retained argument therefore gives S(n + 1) - 1 = (n + 1) squared - 1 = n squared + 2n. For four terms, 3 + 5 + 7 + 9 = 24; the old n-squared formula would give 16.

The reusable contribution is the initial-value and increment argument. It lets the reader obtain the changed sum by identifying the changed range and reusing the already established result. A different progression would require recovering its increment before selecting another formula.