1. Prove that for each n in N, 1+2++ n = n(n+1)/2. 2. Prove that for each n in N, 13 +23+ 3. Prove that for each n in N, 1+3+5+1 4. Prove that for each n ≥ 4,2" -1, then (1+x)" ≥1+nx for each n in N. 11. Prove DeMoivre's Theorem: fort a real number, (cost+i sint)" = cos nt + i sinnt for each n in N, where i = √√-1.
1. Prove that for each n in N, 1+2++ n = n(n+1)/2. 2. Prove that for each n in N, 13 +23+ 3. Prove that for each n in N, 1+3+5+1 4. Prove that for each n ≥ 4,2" -1, then (1+x)" ≥1+nx for each n in N. 11. Prove DeMoivre's Theorem: fort a real number, (cost+i sint)" = cos nt + i sinnt for each n in N, where i = √√-1.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.6: Permutations
Problem 13E
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![1. Prove that for each n in N, 1+2++ n = n(n+1)/2.
2. Prove that for each n in N, 13 +23+
3. Prove that for each n in N, 1+3+5+1
4. Prove that for each n ≥ 4,2" <n!.
+n³ = [n(n+1)/2]².
+(2n-1)= n².
5. Prove that for each n in N, 4 divides 7" - 3".
6. Prove that for each n in N, 5 divides n³ - n.
7. Prove that for each n in N, ✗(-1)+12=(-1)+1
j=1
71
j.
8. Prove that if x1, x2,..., x, are n real numbers in the closed interval [a b].
then x1++ is in the closed interval [a, b].
n
9. Prove that a set with n elements has 2" subsets for each nЄ NU {0}.
10. Prove Bernoulli's inequality: If x > -1, then (1+x)" ≥1+nx for each
n in N.
11. Prove DeMoivre's Theorem: fort a real number,
(cost+i sint)" = cos nt + i sinnt
for each n in N, where i = √√-1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f89dc84-5360-495d-a77a-10dc4ac0ca49%2F30cb9af4-7d1d-4dfd-b0ce-6a9665cd4931%2Fdctieqn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Prove that for each n in N, 1+2++ n = n(n+1)/2.
2. Prove that for each n in N, 13 +23+
3. Prove that for each n in N, 1+3+5+1
4. Prove that for each n ≥ 4,2" <n!.
+n³ = [n(n+1)/2]².
+(2n-1)= n².
5. Prove that for each n in N, 4 divides 7" - 3".
6. Prove that for each n in N, 5 divides n³ - n.
7. Prove that for each n in N, ✗(-1)+12=(-1)+1
j=1
71
j.
8. Prove that if x1, x2,..., x, are n real numbers in the closed interval [a b].
then x1++ is in the closed interval [a, b].
n
9. Prove that a set with n elements has 2" subsets for each nЄ NU {0}.
10. Prove Bernoulli's inequality: If x > -1, then (1+x)" ≥1+nx for each
n in N.
11. Prove DeMoivre's Theorem: fort a real number,
(cost+i sint)" = cos nt + i sinnt
for each n in N, where i = √√-1.
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