prove that 2*34^n-3*23^n +1 is divisible by 726

[esk]

prove that( 2*34^n)-(3*23^n) +1 is divisible by 726

can you prove that( 2*34^n) -(3*23^n) +1 is divisible by 726.

 

[mmm4444bot]

Is this part of some assignment?

If so, what have you tried or thought about thus far? :cool:

 

[esk]

I tried to do it by induction
1 If n=1 then is 0
2 assume 2*34^k-3*23^k+1 is divisible by 726
3 then i have to show that 2*34^k+1-3*23^k+1 +1 is divisible by 726
726=11*11*2*3
and now i'm stuck

 

[lookagain]

I tried to do it by induction
1 If n=1 then is 0

2 assume 2*34^k - 3*23^k + 1 is divisible by 726

3 then i have to show that 2*34^k+1-3*23^k+1 +1 is divisible by 726
No, you must use grouping symbols for this:

2*34^(k + 1) - 3*23^(k + 1) + 1

726=11*11*2*3

and now i'm stuck

...

 

[esk]

i don't know if this is right , but that's what i've done
if n=1, then it works
assume that (2*34^k)-(3*23^k)+1 is divisible by 726 for some random k
then i try to show that (2*34^k+1)-(3*23^k+1)+1 is divisible by 726
(2*34*34^k)-(3*23*23*k)+1=
2*34^k(23+11)-3*23^k(22+1)+1=
2*23*34^k+2*11*34^k-3*22*23^k-3*23^k+1=
2*34^k(22+1)+22*34^k-66*23^k-3*23^k+1=
44*34^k+2*34^k+22*34^K-66*23^k-3*23^k+1+
(2*34^k)-(3*23^k)+1+66(34^k-23^k)=

(2*34^k)-(3*23^k)+1+66(34-23)(34^k-1+...+23^k-1)=

(2*34^k)-(3*23^k)+1+726(34^k-1+...+23^K-1)
is this right ?

 

[lookagain]

i don't know if this is right , but that's what i've done
if n=1, then it works
assume that (2*34^k)-(3*23^k)+1 is divisible by 726 for some random k
then i try to show that (2*34^k+1)-(3*23^k+1)+1 is divisible by 726
(2*34*34^k)-(3*23*23*k)+1=
2*34^k(23+11)-3*23^k(22+1)+1=
2*23*34^k+2*11*34^k-3*22*23^k-3*23^k+1=
2*34^k(22+1)+22*34^k-66*23^k-3*23^k+1=
44*34^k+2*34^k+22*34^K-66*23^k-3*23^k+1 > > + < < This is your typo. This has to be an equals sign instead.
(2*34^k)-(3*23^k)+1+66(34^k-23^k)=

(2*34^k)-(3*23^k)+1+66(34-23)(34^k-1+...+23^k-1)=

(2*34^k)-(3*23^k)+1+726(34^k-1+...+23^K-1)
is this right ?

esk,

do not ignore what I posted to you in post #4 about grouping symbols!


1) Don't vertically cram your lines together.

2) Don't horizontally cram your characters.

3) Use grouping symbols.


---------------------------------------------------------------------

Here could be a versions of yours for increased readability.
I did include redundant/unneeded symbols at times for
more clarification/emphasis:



Assume that [2*(34^k)] - [3*(23^k)] + 1 is divisible by 726 for some random k

then i try to show that [2*34^(k + 1)] - [3*23^(k + 1)] + 1 is divisible by 726

[2*34*(34^k)] - [3*23*(23*k)] + 1 =

[2*(34^k)(23 + 11)] - [3*(23^k)(22 + 1)] + 1 =

2*23*(34^k) + 2*11*(34^k) - 3*22*(23^k) - 3*(23^k) + 1 =

2*(34^k)(22 + 1) + 22*(34^k) - 66*(23^k) - 3*[23^(k + 1)] =

44*(34^k) + 2*(34^k) + 22*(34^k) - 66*(23^k) - 3*[23^(k + 1)] =

2*(34^k) - 3*(23^k) + 1 + 66[(34^k) - (23^k)] =

2*(34^k) - 3*(23^k) + 1 + 66(34 - 23)[34^(k - 1) + ... + 23^(k - 1)] =

2*(34^k) - 3*(23^k) + 1 + 726[34^(k - 1) + ... + 23^(k - 1)]

[/QUOTE]