Chapter 4 Principle of Mathematical Induction Lecture 3

Question 19. n (n + 1) (n + 5) is a multiple of 3.

Question 20. 10^{2n-1}+1 is divisible by 11.

Question 21. x^{2n}-y^{2n} is divisible by x+y .

Question 22. 3^{2n+2}-8n-9 is divisible by 8.

Question 23. 41^n-14^n is a multiple of 27.

Question 16. \frac{1}{1.4}+\frac{1}{4.7}+\frac{1}{7.10}+…+\frac{1}{(3n-2)(3n+1)}=\frac{n}{(3n+1)}

Question 17. \frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+…+\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}

Question 18. 1+2+3+…+n<\frac{1}{8}(2n+1)^2

Question 24. (2n+7)<(n+3)^2

Example 2. 2^n>n

Example 7. 1^2+2^2+…+n^2>\frac{n^3}{3}, n \in N