define a sequence

[dts5044]

define a sequence a_n (n >=1) by setting a_1 = a, a positive integer of your choice, and
a_(n+1) = ...
(a_n)/2 if a_n is even
3a_n +1 if a_n is odd

choose several values for a, and compute enough terms of the corresponding sequence (a_n) (n>=1) until you see a pattern allowing to make a conjecture concerning these sequences. State the conjecture clearly. Can you prove it?

The abstractness of this question confuses me and I honestly cannot find where to start. Can someone help?

 

[daon]

\(\displaystyle .\)

 

[soroban]

Hello, dts5044!

Define a sequence \(\displaystyle a_n\;(n \geq 1)\) by setting \(\displaystyle a_1 = a\), a positive integer of your choice,

\(\displaystyle \text{and: }\;a_{n+1}\; =\;\bigg\{\begin{array}{cc}a_n \div 2 & \text{if } a_n\text{ is even} \\ 3a_n +1 & \text{ if }a_n\text{ is odd} \end{array}\)

Choose several values for \(\displaystyle a\), and compute enough terms of the corresponding sequence
until you see a pattern allowing to make a conjecture concerning these sequences.
State the conjecture clearly. Can you prove it?

Start with any positive integer.
. . If it is even, divide by 2.
. . If it is odd, mutliply by 3 and add 1.
Repeat this process with the new number . . . and so on.


Example: \(\displaystyle a=3\)

. . \(\displaystyle 3 \to 10 \to 5 \to 16 \to 8 \to \underbrace{4 \to 2 \to 1} \to \underbrace{4 \to 2 \to 1} \to \hdots\)
We are stuck in a "loop".


Example: \(\displaystyle a = 85\)

. . \(\displaystyle 85 \to 256 \to 128 \to 64 \to 32 \to 16 \to 8 \to \underbrace{4 \to 2 \to 1} \to \hdots\)


After trying several more starting numbers,
. . we conjecture that all values of \(\displaystyle a\) converge to the 4-2-1 loop.