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Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex
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To prove: The statement is true for all natural numbers using the Extended Principle of Mathematical Induction.
Answer to Problem 33AYU
The statement is true for the natural number terms, it is true for all natural numbers
Explanation of Solution
Given:
Statements says ‘The sumof the interior angles of a convex polygon of
Formula used:
The Extended Principle of Mathematical Induction
Suppose that the following two conditions are satisfied with regard to a statement about natural numbers:
CONDITION I: A statement is true for a natural number
CONDITION II: If the statement is true for some natural number
Proof:
Consider the statement
‘The sumof the interior angles of a convex polygon of
Step 1: Show that statement (1) is true for the natural number
Number of sides = 3. 3-sided polygon is nothing but a triangle.
The sum of the angles of a triangle is
Step 2: Assume that the statement is true for some natural number
That is ‘The sumof the interior angles of a convex polygon of
Step 3: Prove that the statement is true for the next natural number
‘The sumof the interior angles of a convex polygon of
From equation (2), it can be concluded that for
A convex polygon of
‘The sumof the interior angles of a convex polygon of
As the statement is true for the natural number
Chapter 12 Solutions
Precalculus Enhanced with Graphing Utilities
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