Exercise 39 Page 81

Inductive reasoning means we find a pattern for specific cases and then make conjectures for the general case. Let's make a sequence for the sum of the first positive even integers. Next we have to look for a pattern in how the sequence increases, which is called the first difference.

As we can see, the value by which the sequence increases starts at $4$ and then we add $2$ to each increment as we move along the sequence. This means the formula cannot be a linear equation. If the second difference is a constant, we know the formula will be a quadratic equation.

The second difference is a constant, which means this is a quadratic equation of the form $a_n=a{n}^2+bn+c.$ To determine the equation we have to find the values of $a,$ $b,$ and $c.$ Note that $c$ is the sequence value when $n=0.$ Therefore, by going backward from the sequence first number using the established pattern, we can find $c.$

When we know that $c=0,$ we can write the equation. To find the value of $a$ and $b$ we should create a system of equations using two of the ordered pairs. We will use $(1,2)$ and $(2,6).$ Let's solve this system of equations. Having solved for $b,$ we can substitute this value in the first equation to find $a.$ Now that we know the value of $a$ and $b,$ we can write the equation