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n Exercise 12.1 Prove k²= = k=0 72 Exercise 12.2 Prove ³: = k=0 n n(n+1)(2n+1) 6 k=0 n²(n+1)² for n € Z+. for n € Z+. Exercise 12.3 Prove (2k-1)= n² for n € Z+.

n Exercise 12.1 Prove k²= = k=0 72 Exercise 12.2 Prove ³: = k=0 n n(n+1)(2n+1) 6 k=0 n²(n+1)² for n € Z+. for n € Z+. Exercise 12.3 Prove (2k-1)= n² for n € Z+.

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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Currently learning about mathematical induction. I know for induction, we show the base case is true, then assume true for n. Then we need to show the statement is true for n+1.  I can show the base case is true and I know we assume true for n, but I am having trouble showing that n+1 for the following statements?

n Exercise 12.1 Prove k²= = k=0 72 Exercise 12.2 Prove ³: = k=0 k=1 n n Exercise 12.3 Prove Σ (2k-1)= n² for n € Z+. k=0 n(n+1)(2n+1) 6 Exercise 12.4 Prove k*k! = (n + 1)! — 1. k=1 n²(n+1)² Exercise 12.7 Show that is Exercise 12.5 Prove k2k = 2 + (n − 1) 2n+¹. - Exercise 12.8 Show that for n € Z+. Exercise 12.6 Find and prove the correctness of a formula for 2k for ne Zt. k=0 (2n)! 2n for n € Z+. (3n)! 6″ an integer for n € Z+. n³-n+12 6 an integer for ne Zt. 23n+3n+2 5 Exercise 12.9 Show that is an integer for ne Zt. Exercise 12.10 Show that is 2³-22 an integer for n € Z+. 7 Exercise 12.11 Show that is an integer for ne Zt. n Exercise 12.12 Show that is 5+2+9+10 4 Exercise 12.13 Show that n! ≥ 2" for integers n ≥ 4. Exercise 12.14 Show that (2n)!> 32n+1 for integers n ≥ 4. an integer for n € Z+.
Transcribed Image Text:n Exercise 12.1 Prove k²= = k=0 72 Exercise 12.2 Prove ³: = k=0 k=1 n n Exercise 12.3 Prove Σ (2k-1)= n² for n € Z+. k=0 n(n+1)(2n+1) 6 Exercise 12.4 Prove k*k! = (n + 1)! — 1. k=1 n²(n+1)² Exercise 12.7 Show that is Exercise 12.5 Prove k2k = 2 + (n − 1) 2n+¹. - Exercise 12.8 Show that for n € Z+. Exercise 12.6 Find and prove the correctness of a formula for 2k for ne Zt. k=0 (2n)! 2n for n € Z+. (3n)! 6″ an integer for n € Z+. n³-n+12 6 an integer for ne Zt. 23n+3n+2 5 Exercise 12.9 Show that is an integer for ne Zt. Exercise 12.10 Show that is 2³-22 an integer for n € Z+. 7 Exercise 12.11 Show that is an integer for ne Zt. n Exercise 12.12 Show that is 5+2+9+10 4 Exercise 12.13 Show that n! ≥ 2" for integers n ≥ 4. Exercise 12.14 Show that (2n)!> 32n+1 for integers n ≥ 4. an integer for n € Z+.
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