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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
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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