Chapter 4 Principle of Mathematical Induction Lecture 3

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

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