Question

3x-1
------- = 11
4​

Answer 1

Answer:

$\huge \color{black} \boxed{ \colorbox{yellow}{Aɴsᴡᴇʀシ︎}}$

The 3x+1 problem concerns an iterated function and the question of whether it always reaches 1 when starting from any positive integer. It is also known as the Collatz problem or the hailstone problem.

The function is

For example, {\displaystyle f(3)=10}{\displaystyle f(3)=10}. And {\displaystyle f(10)=5}{\displaystyle f(10)=5}. This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, ... which indeed reaches 1. A sequence obtained by iterating the function from a given starting value is sometimes called "the trajectory" of that starting value.

Obviously there can be no consecutive odd numbers in any trajectory, but there may certainly be consecutive even numbers, especially when the trajectory reaches a power of 4, in which the trajectory quickly plummets to 1 after passing through all the intervening powers of 2. (Note that since the odd-indexed powers of 2 are congruent to 2 modulo 3, they are only reachable from halving a power of 4).

Answer 2

Answer:

$</p><p>\huge \color{black} \boxed{ \colorbox{yellow}{Aɴsᴡᴇʀシ︎}}$

The 3x+1 problem concerns an iterated function and the question of whether it always reaches 1 when starting from any positive integer. It is also known as the Collatz problem or the hailstone problem.

The function is

For example, {\displaystyle f(3)=10}{\displaystyle f(3)=10}. And {\displaystyle f(10)=5}{\displaystyle f(10)=5}. This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, ... which indeed reaches 1. A sequence obtained by iterating the function from a given starting value is sometimes called "the trajectory" of that starting value.