Order 9 of the following sentences so that they form a proof by induction of the statement: 1^2+2^2+3^2+...+𝑛2=𝑛(𝑛+1)(2𝑛+1)/6 for all natural numbers n.

Question

Order 9 of the following sentences so that they form a proof by induction of the statement:

1²+2²+3²+...+𝑛²=𝑛(𝑛+1)(2𝑛+1)/6 for all natural numbers n.

Options

1. We will proceed by induction to prove that 1²+2²+3²+...+𝑛²=𝑛(𝑛+1)(2𝑛+1)/6 for all natural numbers n.
2. Base Case: n=1.
3. LHS: 1²=1, RHS: 1(2)(3)/6=1, and 1=1.
4. Inductive Hypothesis: Assume 1²+2²+3²+. . .+k²=k(k+1)(2k+1)/6 for some natural numbers k.
5. 1²+2²+3²+. . .+k²+(k+1)²=k(k+1)(2k+1)/6+(k+1)²,
6. And (k+1)[k²/3+k/6+k+1]=(k+1)/6[2k²+7k+6]=(k+1)(k+2)(2k+3)/6.
7. Which is equal to (k+1)[(k(2k+1)/6+(k+1)]=(k+1)[k²/3+k/6+k+1].
8. Which is finally equal to (k+1)((k+1)+1)(2(k+1)+1)/6.
9. Then by mathematical induction, 1²+2²+3²+...+𝑛²=n(n+1)(2n+1)/6 for all natural numbers n.
10. Inductive Hypothesis: Assume 1²+2²+3²+. . .+n²=n(n+1)(2n+1)/6 for all natural numbers n.
11. Base Case: n=0
12. Inductive Hypothesis: Assume 1²+2²+3²+. . .+k²=k(k+1)(2k+1)/6
13. LHS: 0²=0, RHS: 0(1)(1)/6=0, so 0=0

Answer 1

1, 2, 3, 12, 5, 6, 7, 8, 9