Standard +0.8 This is a proof question requiring students to express the rth odd number as (2r-1), cube it, expand to get 8r³-12r²+6r-1, then apply standard summation formulas and simplify algebraically to reach the given result. It requires multiple steps of algebraic manipulation and careful bookkeeping, going beyond routine exercises, but uses only standard techniques that Further Maths students should know well.
In this question you must show detailed reasoning.
In this question you may assume the results for
$$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$
Show that the sum of the cubes of the first \(n\) positive odd numbers is
$$n^2(2n^2 - 1).$$ [5]
In this question you must show detailed reasoning.
In this question you may assume the results for
$$\sum_{r=1}^{n} r^3, \quad \sum_{r=1}^{n} r^2 \quad \text{and} \quad \sum_{r=1}^{n} r.$$
Show that the sum of the cubes of the first $n$ positive odd numbers is
$$n^2(2n^2 - 1).$$ [5]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q6 [5]}}