Invariance Math Example 2
Follow the full solution, then compare it with the other examples linked below.
Example 2
mediumA sequence starts at 1 and each step either doubles the value or adds 3. Show that the parity (odd/even) of the value changes predictably and identify an invariant.
Solution
- 1 Start with 1 (odd). Doubling an odd number gives an even number. Adding 3 to an odd number gives an even number.
- 2 From an even number: doubling gives even; adding 3 gives odd.
- 3 Observe: starting odd, after one step we are always even. From even, we may go to odd or even.
- 4 Invariant: after any 'double' step, the result is always even. This parity behaviour is preserved regardless of how many steps we take.
Answer
Invariants are quantities or properties that remain unchanged (or change in a predictable, rule-governed way) across a process. Identifying invariants often unlocks the analysis of a dynamic system.
About Invariance
A property of a mathematical object that remains unchanged when the object undergoes a particular transformation or operation.
Learn more about Invariance âMore Invariance Examples
Example 1 easy
Show that the sum of the digits of a multiple of 9 is always a multiple of 9. Verify with [formula]
Example 3 easyShow that the expression [formula] is invariant under the transformation [formula].
Example 4 mediumIn a game, you start with the number 6. Each move you may subtract 1 or divide by 2 (if even). Show