Patterns Inductive Reasoning

Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment - Euclidean geometry set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer.

The introduction of coordinates by Rene Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of calculus in the 17^th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

Patterns

A pattern constitutes a set of numbers or objects in which all the members are related to each other by a specific rule. Pattern is also a sequence. There can be finite or infinite number of members in a pattern.

Eg : 1,4,9,....... This is an example for a pattern . In this pattern the relation is squares of natural numbers. That is 1 is square of 1 itself. Then square of 2 = 4 , square of 3 = 9 and so on.

So in a pattern there will be three related terms. They are Number, Rule and Sequence.

Induction

Definition of Induction

• Mathematical Induction is a method generally used to prove or establish that a given statement is true for all natural numbers.
• It is a method to prove a proposition which is valid for infinitely many different values of a variable.

Example of Induction

• Mathematical induction can be used to prove that 1 + 3 + 5 + ---- + (2n - 1) = n2 for all positive integers.

  Let Pn be the statement 1 + 3 + 5 + ---- + (2n - 1) = n².

  P₁ is true because (2(1) - 1) = 12.

  Assume that Pk is true, so that Pk: 1 + 3 + 5 + ------ + (2k - 1)

  = k². [The Inductive Hypothesis.]

  The next term on the left hand side would be [2(k + 1) - 1] = (2k + 1).

  Add (2k + 1) on both sides to Pk.]

  1 + 3 + 5 + ------ + (2k - 1) + (2k + 1) = k² + (2k + 1)

  = (k + 1)² = P_k+1

  So, the equation is true for n = k + 1.

  Therefore, Pn is true for all positive integers, by mathematical induction.