Use the Principle of Mathematical Induction in the following Exercises. 17. A sequence al, a2, аз numbers k22. Show that a3.7H for all natural numbers. is defined by letting al-3 and a,-7aH for all natural 18. A sequence bo b,b, is defined by letting bo- 5 and b.4+b.1 for all natural numbers k. Show that b54n for all natural number n using mathematical induction 19. A sequence d, d,, d, ...is defined by letting d 2 and d-for all natural numbers, k22. Show that d- for all n E N 7n 20. Prove that for all n E N cos α+cos (α + β) + cos (α + 2B) + cos (α + (n-1) β) sin sin 2 e 2" sin θ sín n θ . 21. Prove that, cos θ cos 2θ cos29 cos2-e , for a all n e N (n+1) sin 22, Prove that, sinθ+sin 2θ+sin 3θ + + sinn®- for alln E N sin 23. Show thatis a natural number for all neE N. 5 3 15 1 13 n+1 n+2 2n 24 24. Prove that, for all natural numbers >1 25. Prove that number of subsets of a set containing n distinct elements is 2, for all
Use the Principle of Mathematical Induction in the following Exercises. 17. A sequence al, a2, аз numbers k22. Show that a3.7H for all natural numbers. is defined by letting al-3 and a,-7aH for all natural 18. A sequence bo b,b, is defined by letting bo- 5 and b.4+b.1 for all natural numbers k. Show that b54n for all natural number n using mathematical induction 19. A sequence d, d,, d, ...is defined by letting d 2 and d-for all natural numbers, k22. Show that d- for all n E N 7n 20. Prove that for all n E N cos α+cos (α + β) + cos (α + 2B) + cos (α + (n-1) β) sin sin 2 e 2" sin θ sín n θ . 21. Prove that, cos θ cos 2θ cos29 cos2-e , for a all n e N (n+1) sin 22, Prove that, sinθ+sin 2θ+sin 3θ + + sinn®- for alln E N sin 23. Show thatis a natural number for all neE N. 5 3 15 1 13 n+1 n+2 2n 24 24. Prove that, for all natural numbers >1 25. Prove that number of subsets of a set containing n distinct elements is 2, for all
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use
induction on to prove...
Related questions
Question
Hi, please help me with #21 in the attached image.
Expert Solution
Trending now
This is a popular solution!
Step by step
Solved in 10 steps with 10 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning