9. Use mathematical lowing inequalities. induction to establish each of the fol- (a) [BB] 2">n², for n ≥ 5. for all n ≥ −3. (b) 2" (c) [BB] n! > n³ for all n ≥ 6. (d) (1 + )" ≥ 1+1, for n € N. (e) For any x € R, x > -1, (1+x)" ≥ 1+nx for all nEN.
9. Use mathematical lowing inequalities. induction to establish each of the fol- (a) [BB] 2">n², for n ≥ 5. for all n ≥ −3. (b) 2" (c) [BB] n! > n³ for all n ≥ 6. (d) (1 + )" ≥ 1+1, for n € N. (e) For any x € R, x > -1, (1+x)" ≥ 1+nx for all nEN.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Q9 e only and Q10 c only. Thanks
![i=1
induction to establish each of the fol-
9. Use mathematical
lowing inequalities.
(a) [BB] 2"> n², for n ≥ 5.
(b) 2 ≥ for all n ≥ −3.
(c) [BB] n! > n3³ for all n ≥ 6.
(d) (1+)" ≥ 1+1, for n € N.
2
(e) For any x ER, x>-1, (1+x)" ≥ 1+nx for all
neN.
(f) For any integer n ≥ 2, +1+2+3+...
+...+ ½ /n
13
24
1
1
1
1
(g)
1
32 +...+
<2-- for all n ≥ 2.
22
n
n
(h)
> √n for n ≥ 2.
S
i=1
(i) [BB] 1(3) (5)
(2n-1) ≥ 2(4) (6)
(2n-2) for
every integer n ≥ 2.
10. Suppose c, X1, X2, ..., Xn, Yı, Y2, ..., yn are 2n+1 given
numbers. Prove each of the following assertions by math-
ematical induction.
n
n
n
(a) [BB] Σ(x + y) = Σx + Σy for n ≥ 1,
i=1
i=1
i=1
n
n
(b) Σexi = c Σx for n ≥ 1.
i=1
i=1
n
(c) Σ(x₁ - x₁-1) = x₁ - x₁ for n ≥ 2.
xn
i=2
11 IRRUE
h
following "proof" that in an
e e
+
-
+
n2
●
A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16cd84dd-3536-4a60-93ca-72803276df73%2F508628c6-459b-4a03-bf0b-1879ac75d7de%2Fkbb19kd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:i=1
induction to establish each of the fol-
9. Use mathematical
lowing inequalities.
(a) [BB] 2"> n², for n ≥ 5.
(b) 2 ≥ for all n ≥ −3.
(c) [BB] n! > n3³ for all n ≥ 6.
(d) (1+)" ≥ 1+1, for n € N.
2
(e) For any x ER, x>-1, (1+x)" ≥ 1+nx for all
neN.
(f) For any integer n ≥ 2, +1+2+3+...
+...+ ½ /n
13
24
1
1
1
1
(g)
1
32 +...+
<2-- for all n ≥ 2.
22
n
n
(h)
> √n for n ≥ 2.
S
i=1
(i) [BB] 1(3) (5)
(2n-1) ≥ 2(4) (6)
(2n-2) for
every integer n ≥ 2.
10. Suppose c, X1, X2, ..., Xn, Yı, Y2, ..., yn are 2n+1 given
numbers. Prove each of the following assertions by math-
ematical induction.
n
n
n
(a) [BB] Σ(x + y) = Σx + Σy for n ≥ 1,
i=1
i=1
i=1
n
n
(b) Σexi = c Σx for n ≥ 1.
i=1
i=1
n
(c) Σ(x₁ - x₁-1) = x₁ - x₁ for n ≥ 2.
xn
i=2
11 IRRUE
h
following "proof" that in an
e e
+
-
+
n2
●
A
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