Theory of Numbers (VII): Principle of Mathematical Induction: Fibonacci Number: Prove that F1 + F2 + F3 +…+ Fn = F(n + 2) – 1

Theory of Numbers (VII) Principle of Mathematical Induction Fibonacci Number Suppose that F1 = 1, F2 = 1, F3 =1, F4 = 3, F5 = 5, and in general Fn = Fn-1 + Fn-2 for n ≥ 3 ( Fn is called the nth Fibonacci number.) Prove that F1 + F2 + F3 +…+ Fn = F(n + 2) – 1

Theory of Numbers (VIII): Principle of Mathematical Induction: Fibonacci Number: Prove that F1 + F3 + F5 +…+ F(2n – 1) = F2n

Theory of Numbers (VIII) Principle of Mathematical Induction Fibonacci Number Prove that F1 + F3 + F5 +…+ F(2n – 1) = F2n

Theory of Numbers (IX): Principle of Mathematical Induction: Fibonacci Number: Prove that F2 + F4 + F6 +…+ F2n = F(2n+1) – 1

Theory of Numbers (IX) Principle of Mathematical Induction Fibonacci Number Prove that F2 + F4 + F6 +…+ F2n = F(2n+1) – 1

Divisibility Rules for the numbers from 2 to 20

Derive rules to test whether a number is divisible by N, where N ranges from 2 to 20. E.g. A number is divisible by 3 if the sum of the digits is divisible by 3. Show that a palindromic number which has an even number of digits is always divisible by 11.

Math Theory

Can any set that is not a group (Z for example) still be a ring or is it necessary that a set must be a group to be a ring? Please give an example and counter example.