For n=1, the left side is 1(2)=2 and the right side is 1(2)(3)/3=2.
Assume the result holds for a positive integer k: r=1∑kr(r+1)=3k(k+1)(k+2).
Then r=1∑k+1r(r+1)=3k(k+1)(k+2)+(k+1)(k+2)=(k+1)(k+2)(3k+1)=3(k+1)(k+2)(k+3). This is the required expression for n=k+1. The base case and inductive step establish the statement for every positive integer n.
Verifies the base case.
[1 mark]
States the induction assumption.
[1 mark]
Adds the correct next term.
[1 mark]
Simplifies to the required expression for k+1.
[1 mark]
Makes a valid induction conclusion.
[1 mark]
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.