MHB Proving $T(2^k)=2^k \cdot k$ with Induction

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The discussion focuses on using mathematical induction to prove that for powers of 2, the recurrence relation T(n) satisfies T(2^k) = 2^k * k. The proof begins with a base case for k=1, confirming T(2) equals 2. The induction step assumes the hypothesis holds for k and demonstrates it for k+1, leading to the conclusion that T(2^k) holds for all natural numbers k. Participants suggest improving the clarity of the initial statement to better frame the goal of the proof. Overall, the formulation is deemed correct and clear with minor suggestions for enhancement.
evinda
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Hello! (Wave)

Use induction to show that if $n$ is a power of $2$ then the solution of the recurrence relation$$\left\{\begin{matrix}
2 &, \text{ if } n=2 \\
2T\left( \frac{n}{2} \right )+n & , \text{ if } n=2^k, \text{ for } k>1
\end{matrix}\right.$$

is $T(n)=n \lg n$.That's what I have tried:

We will use induction on $k$ to show that $T(2^k)=2^k \cdot k$.

For $k=1 \Rightarrow n=2^1=2$ we have: $T(2)=2=2 \lg 2 \checkmark$.

We suppose that for some $k \geq 1$ it holds that $T(2^k)=2^k \cdot k$.

For $k+1$: $T(2^{k+1})=2T\left( \frac{2^{k+1}}{2} \right )+2^{k+1}=2T(2^k)+2^{k+1} \overset{\text{induction hypothesis}}{=}2 \cdot 2^k \cdot k+2^{k+1}=2^{k+1} \cdot k+2^{k+1}=2^{k+1}(k+1)$.Thus, $\forall k \in \mathbb{N}$ it holds that $T(2^k)=2^k \cdot k$.Could you tell me if my formulation is right or if I could improve something? (Thinking)
 
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evinda said:
Hello! (Wave)

Use induction to show that if $n$ is a power of $2$ then the solution of the recurrence relation$$\left\{\begin{matrix}
2 &, \text{ if } n=2 \\
2T\left( \frac{n}{2} \right )+n & , \text{ if } n=2^k, \text{ for } k>1
\end{matrix}\right.$$

is $T(n)=n \lg n$.That's what I have tried:

We will use induction on $k$ to show that $T(2^k)=2^k \cdot k$.

For $k=1 \Rightarrow n=2^1=2$ we have: $T(2)=2=2 \lg 2 \checkmark$.

We suppose that for some $k \geq 1$ it holds that $T(2^k)=2^k \cdot k$.

For $k+1$: $T(2^{k+1})=2T\left( \frac{2^{k+1}}{2} \right )+2^{k+1}=2T(2^k)+2^{k+1} \overset{\text{induction hypothesis}}{=}2 \cdot 2^k \cdot k+2^{k+1}=2^{k+1} \cdot k+2^{k+1}=2^{k+1}(k+1)$.Thus, $\forall k \in \mathbb{N}$ it holds that $T(2^k)=2^k \cdot k$.Could you tell me if my formulation is right or if I could improve something? (Thinking)

Looks fine for me :)
 
ZaidAlyafey said:
Looks fine for me :)

Would it be better to say at the beginning something like that?

We want to show that when $n$ is some power of $2$ then the solution of the recurrence relation is $T(n)=n \lg n$, i.e. that for $n=2^k$ where $k$ is some positive integer the solution is equal to $2^k \cdot k$.
 
Last edited:
evinda said:
Would it be better to say at the beginning something like that?

We want to show that when $n$ is some power of $2$ then the solution of the recurrence relation is $T(n)=n \lg n$, i.e. that for $n=2^k$ where $k$ is some positive integer the solution is equal to $2^k \cdot k$.

Yeah, it is better.
 
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