Explanation Given: The series of sketches starts with an equilateral triangle having sides of length 1 . In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is 1 3 the length of the sides of the preceding triangles. Develop a formula for the number of sides of the n t h figure. Use mathematical induction to prove the answer. Given that the series of sketches starts with an equilateral triangle having sides of length . In the following steps, equilateral triangles are constructed on each side of the preceding figure. The length of the sides of each new triangle is 1 3 the length of the sides of the preceding triangles. The first triangle has 3 sides. The second figure contains 12 sides. The third figure contains 48 sides. The fourth figure contains 128 sides of a triangle. So the series should be as follows: 3 , 12 , 48 , 192 , ... .The series can be rewritten as: 3 ( 4 ) 0 , 3 ( 4 ) 1 , 3 ( 4 ) 3 , 3 ( 4 ) 4 , ..........3 ( 4 ) n − 1 where n is a natural number. It can be rewritten as 3 [ 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 ] . The sum of the geometric series: 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 4 − 1 = 4 n − 1 3 . Thereby our given series becomes: 3 [ 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 ] = 3 [ 4 n − 1 3 ] . On rewriting this equation we can obtain the statement which is to be proved, P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 By Proof of Induction, in the first step we have to prove P ( n ) is true for n = 1 . In the second step is we have to consider P ( n ) is true for n = k , where k is any value. It means P ( k ) is true. This is called Induction Hypothesis. We will now prove directly that P ( k ) ⇒ P ( k + 1 ) , which means P ( k + 1 ) is true by using the fact that P ( k ) is true. This is called Induction Step. The combination of these steps shows that P ( n ) is true for all values of n. Formula used: P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 . Calculation: Description P ( n ) : 1 + 4 + 16 + 64 + ........... + ( 4 ) n − 1 = 4 n − 1 3 Step 1: To prove P ( 1 ) is true...

College Algebra with Modeling & Visualization (5th Edition)

5th Edition

ISBN: 9780321826138

Author: Gary K. Rockswold

Publisher: PEARSON

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Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

Chapter 8.5, Problem 30E

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Formula for the number of sides of the $n^{t h}$ figure. Use mathematical induction to prove the answer.

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Chapter 8.5, Problem 29E

Chapter 8.5, Problem 31E

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Chapter 8 Solutions

College Algebra with Modeling & Visualization (5th Edition)

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

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Formula for the number of sides of the n t h figure. Use mathematical induction to prove the answer.

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Formula for the number of sides of the nth figure. Use mathematical induction to prove the answer.

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Formula for the number of sides of the $n^{t h}$ figure. Use mathematical induction to prove the answer.