Exercise 63 Page 714

We need to prove that the following statement is true for all positive integers. 2 + 6 + 12 + ⋯ + n(n+ 1) = n(n+1)(n+2)/3 To do it, we will use mathematical induction.

Step 1

We begin by proving that the statement is true for n=1. When n= 1, the left-hand side becomes 1( 1+1) = 2. Let's compute the right-hand side. 1( 1+1)( 1+2)/3 = 1(2)(3)/3 = 2 ✓ Since both sides are equal, we have that the statement is true for n=1.

Step 2

Let's assume that the statement is true for a natural number k. Inductive Hypothesis: 2 + 6 + 12 + ⋯ + k(k+ 1) = k(k+1)(k+2)/3

Step 3

Here, we have to prove that the given statement is true for n=k+1. Let's see how the left- and right-hand sides change when we substitute n= k+1.

Left-hand side	Right-hand side
2 + 6 + 12 + ⋯ + ( k+1)( k+1+1)	( k+1)( k+1+1)( k+1+2)/3
2 + 6 + 12 + ⋯ + (k+1)(k+2)	(k+1)(k+2)(k+3)/3

To verify that both sides are equal, we will start with inductive hypothesis and perform some operations. Notice that the final equation is the one we wanted to prove. As such, given statement is true for n=k+1, and so it is true for all positive integers.