2. Inductive and Deductive Reasoning
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Is the formula going to be a linear equation or a quadratic equation? How can you find out?
a_n=n(n+1)
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.
(I): LHS-b=RHS-b
(II): a= 2-b
(II): Distribute 4
(II): Add terms
(II): LHS-8=RHS-8
(II): .LHS /(- 2).=.RHS /(- 2).
(II): Rearrange equation
(I): b= 1
(I): Subtract term
(I), (II): Rearrange equation