A journey of a thousand miles begins with a single step. – Lao Tzu
| Exercise 4.1 |
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