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

Question

Hi, please help me with #21 in the attached image.

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
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Transcribed Image Text

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

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