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