1. ## Proof of Induction

It is given that$\displaystyle sn=2*2+3*2^2+4*2^2+........ +(n+1)*2^n$

a) By first considering the values of $\displaystyle \frac{S1}{1}$,$\displaystyle \frac{S2}{2}$,$\displaystyle \frac{S3}{3}$,$\displaystyle \frac{S4}{4}$ and making a conjecture for a formula for $\displaystyle \frac{Sn}{n}$, make a conjecture for a formula for Sn.

b) Use induction to prove that your conjecture is correct.

For a) i did $\displaystyle S1=4, S2=6, S3=8$

so the conjecture for $\displaystyle Sn=2n+2$

b) $\displaystyle S1=4$, $\displaystyle S1=2*1+2=4$
so n=1 is true.

$\displaystyle Sk=2k+2$
$\displaystyle Sk+1=2(k+1)+2 = 2k+4$

what do i do next? or how to i prove $\displaystyle Sk+1=2k+4$

thanks.

2. Originally Posted by BabyMilo
It is given that$\displaystyle sn=2*2+3*2^2+4*2^2+........ +(n+1)*2^n$

a) By first considering the values of $\displaystyle \frac{S1}{1}$,$\displaystyle \frac{S2}{2}$,$\displaystyle \frac{S3}{3}$,$\displaystyle \frac{S4}{4}$ and making a conjecture for a formula for $\displaystyle \frac{Sn}{n}$, make a conjecture for a formula for Sn.

b) Use induction to prove that your conjecture is correct.

For a) i did $\displaystyle S1=4, S2=6, S3=8$
No, S2= 2(2)+ 3(4)= 4+ 12= 16, not 6 and S3= 2(2)+ 3(4)+ 4(8)= 4+ 12+ 32= 48, not 8.

Why not do what you were told to do- look at Sn/n first?

S1/1= 4, S2/2= 8, S3/3= 48/3= 16, ...

so the conjecture for $\displaystyle Sn=2n+2$

b) $\displaystyle S1=4$, $\displaystyle S1=2*1+2=4$
so n=1 is true.

$\displaystyle Sk=2k+2$
$\displaystyle Sk+1=2(k+1)+2 = 2k+4$

what do i do next? or how to i prove $\displaystyle Sk+1=2k+4$

thanks.

3. Originally Posted by HallsofIvy
No, S2= 2(2)+ 3(4)= 4+ 12= 16, not 6 and S3= 2(2)+ 3(4)+ 4(8)= 4+ 12+ 32= 48, not 8.

Why not do what you were told to do- look at Sn/n first?

S1/1= 4, S2/2= 8, S3/3= 48/3= 16, ...
so i have $\displaystyle \frac{Sn}{n} = 2^{n+1}$

$\displaystyle \rightarrow Sn=2^{n+1}*n$

4. Originally Posted by BabyMilo
so i have $\displaystyle \frac{Sn}{n} = 2^{n+1}$

$\displaystyle \rightarrow Sn=2^{n+1}*n$
now im stuck on

$\displaystyle 2^{k+2}+2(2^{k+1}*k) = 2^{k+2}*(k+1)$

5. Originally Posted by BabyMilo
now im stuck on

$\displaystyle 2^{k+2}+2(2^{k+1}*k) = 2^{k+2}*(k+1)$
Hi BabyMilo,

P(k)

$\displaystyle S_k=k2^{k+1}$

then we try to express P(k+1) in terms of P(k) to find out if there is a cause

P(k+1)

$\displaystyle S_{k+1}=(k+1)2^{k+2}$

Let's see if this is what we get when we add the (k+1)th term to $\displaystyle S_k$

$\displaystyle S_k+T_{k+1}=k2^{k+1}+(k+2)2^{k+1}$

since the (k+1)th term is $\displaystyle (k+1+1)2^{k+1}$

$\displaystyle S_k+T_{k+1}=2^{k+1}(k+k+2)=2(k+1)2^{k+1}=(k+1)2^{k +2}$

As $\displaystyle S_k$ being valid will cause $\displaystyle S_{k+1}$ to be valid,

there is a cause and effect relationship, so you only need test the first one.

6. Originally Posted by Archie Meade
Hi BabyMilo,

P(k)

$\displaystyle S_k=k2^{k+1}$

then we try to express P(k+1) in terms of P(k) to find out if there is a cause

P(k+1)

$\displaystyle S_{k+1}=(k+1)2^{k+2}$

Let's see if this is what we get when we add the (k+1)th term to $\displaystyle S_k$

$\displaystyle S_k+T_{k+1}=k2^{k+1}+(k+2)2^{k+1}$

since the (k+1)th term is $\displaystyle (k+1)2^{k+1+1}$

$\displaystyle S_k+T_{k+1}=2^{k+1}(k+k+2)=2(k+1)2^{k+1}=(k+1)2^{k +2}$

As $\displaystyle S_k$ being valid will cause $\displaystyle S_{k+1}$ to be valid,

there is a cause and effect relationship, so you only need test the first one.
Can you show me how you got from to $\displaystyle S_k+T_{k+1}=2^{k+1}(k+k+2)$

7. Originally Posted by BabyMilo
Can you show me how you got from to $\displaystyle S_k+T_{k+1}=2^{k+1}(k+k+2)$
Yes,

if we take $\displaystyle 2^{k+1}$ as the common factor of both terms

$\displaystyle 2^{k+1}[k]+2^{k+1}(k+2)=2^{k+1}\left([k]+(k+2)\right)$