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

Lecture - 3 Chapter 4 Principle of Mathematical Induction

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