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.
2&=2
2+4&=6
2+4+6&=12
2+4+6+8&=20
2+4+6+8+10&=30
2+4+6+8+10+12&=42
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=an^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.
a_n=an^2+bn ⇔ a_n=n(an+b)
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).
2= 1(a( 1)+b) 6= 2(a( 2)+b)
⇒ 2=a+b 6=4a+2b
Let's solve this system of equations.
