Inductive Reasoning is reason based on patterns, as opposed to rules, which is Deductive Reasoning. Inductive Reasoning provides conjectures, which are conclusions based on inductive reasoning.
Geometry includes making conjectures and conducting proofs, which involve making deductions through a series of observational steps. Thus, Inductive Reasoning is very important to understand before beginning Geometry.
Here are some examples:
(1) Make a conjecture about the $ n$th term in the following sequence:
| $ n$ | $ 1$ | $ 2$ | $ 3$ | $ 4$ | $ 5$ | … | $ n$ |
| $ {{a}_{n}}$ | $ -2$ | $ 0$ | $ 2$ | $ 4$ | $ 6$ | … | $ ?$ |
There are several ways to determine the next term, some of which were shown in the Sequence and Series section.
Let’s use common sense. Note that the first term is $ -2$, with the next term $ 2$ more than the previous term (the common difference). This is a linear relationship, since the difference between terms is constant ($ 2$).
To see how to define the next term in terms of $ n$, notice each term is $ 4$ less than the twice the term number, so, for the $ n$th term, it would be $ {{a}_{n}}=2n-4$. Try it; it works!
(2) Make a conjecture about the $ n$th term in the following sequence:
| $ n$ | $ 1$ | $ 2$ | $ 3$ | $ 4$ | $ 5$ | … | $ n$ |
| $ {{a}_{n}}$ | $ 2$ | $ 5$ | $ 10$ | $ 17$ | $ 26$ | … | $ ?$ |
Notice:
           Pattern:     $2$     $ 5$     $ 10$    $ 17$    $ 26$
      1st difference:   $ 3$     $ 5$     $ 7$     $ 9$
  2nd difference:     $ 2$     $ 2$     $ 2$
This one’s a little more complicated, since the first differences of  aren’t constant ($ 3,5,7,9,…$) . But notice that the second differences are ($ 2,2,2,2,…$)! Thus, the sequence has a Quadratic relationship.
There are several ways to determine what the quadratic relationship is. One way is to perform a Regression Analysis using the data. Another is to observe that each successive term is the square of the previous term, plus $ 1$. Thus, the sequence is $ {{a}_{n}}={{n}^{2}}+1$.
(3) Find the next set of dots in the following pattern:
Of course, we could add the dots around the last figure to see there would be $ 5$ more dots, say, on the right-hand side. There would be $ 15$ dots in the figure.
But also notice:
         Pattern:     $1$     $ 3$     $ 6$    $ 10$
      1st difference:   $ 2$     $ 3$     $ 4$
  2nd difference:     $ 1$     $ 1$
Since the second differences are constant, this is also a quadratic.
Here’s a neat trick. If we multiply the number of dots by $ 2$ for the pattern, we get $ 2,6,12,20$ for $ n=1,2,3,4$. This is $ 1\times 2,\,\,\,2\times 3,\,\,\,3\times 4,\,\,\,4\times 5$, or $ n\left( {n+1} \right)$. Divide by $ 2$ and we get our original pattern of $ 1,3,6,10$. Thus, the pattern for this sequence is $ \displaystyle \frac{{n\left( {n+1} \right)}}{2}$, where $ n$ is the placeholder in the pattern. Try it; it works!
(4) Draw the next shape in this pattern:
These are fun; we can make a conjecture about the next shape. It must be $ 5$ by $ 5$, and have a heart in the lower left-hand corner:
On to Points, Lines and Planes!