Describe how mathematical induction could be adapted to prove a statement for all integers less than or equal to some integer n, rather than for integers greater than or equal to s... Describe how mathematical induction could be adapted to prove a statement for all integers less than or equal to some integer n, rather than for integers greater than or equal to some base case. What adjustments would need to be made to the standard inductive proof structure?
Understand the Problem
The question asks how to adapt mathematical induction to prove a statement for all integers less than or equal to some integer n, instead of the usual case of proving it for integers greater than or equal to a base case. It also asks about the necessary adjustments to the standard inductive proof structure.
Answer
To prove $P(m)$ for all integers $m \le n$ using induction: 1. Prove $P(n)$. 2. Assume $P(k)$ for $k \le n$. 3. Prove $P(k) \implies P(k-1)$. 4. Conclude $P(m)$ for all $m \le n$.
Answer for screen readers
To adapt mathematical induction to prove a statement $P(m)$ for all integers $m \le n$:
- Base Case: Prove that $P(n)$ is true.
- Inductive Hypothesis: Assume that $P(k)$ is true for some integer $k \le n$.
- Inductive Step: Prove that $P(k-1)$ is true, assuming that $P(k)$ is true (i.e., show that $P(k) \implies P(k-1)$).
- Conclusion: Conclude that $P(m)$ is true for all integers $m \le n$.
Steps to Solve
- Base Case Adjustment
Instead of proving the statement for a smallest integer, you prove it for the largest integer $n$ for which you want to show the statement holds.
- Inductive Hypothesis Adjustment
Assume that the statement holds for some integer $k$ such that $k \le n$.
- Inductive Step Adjustment
You need to show that if the statement holds for $k$, then it also holds for $k-1$. This is the key difference: you're proving the statement holds for a smaller integer, rather than a larger one. Thus, you will show that $P(k) \implies P(k-1)$.
- Conclusion
Conclude that the statement holds for all integers less than or equal to $n$.
To adapt mathematical induction to prove a statement $P(m)$ for all integers $m \le n$:
- Base Case: Prove that $P(n)$ is true.
- Inductive Hypothesis: Assume that $P(k)$ is true for some integer $k \le n$.
- Inductive Step: Prove that $P(k-1)$ is true, assuming that $P(k)$ is true (i.e., show that $P(k) \implies P(k-1)$).
- Conclusion: Conclude that $P(m)$ is true for all integers $m \le n$.
More Information
This is a variation of mathematical induction. Standard induction proves a statement for integers greater than or equal to a base case, while this adapted version proves it for integers less than or equal to a base case. This adjustment is useful when the logical flow of the proof is more naturally directed from larger to smaller values.
Tips
A common mistake is to try to apply the standard inductive step (proving $P(k+1)$ from $P(k)$) when you need to prove $P(k-1)$ from $P(k)$. For this variation of induction, you must work "downwards" from $n$, not "upwards" from some base case less than $n$.
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