mathematical induction proof problem

A

allegansveritatem

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Here is what I'm being asked to prove for every positive integer n:
proof 01-19 2.PNG

First I tested with n=1:
proof01-19.PNG

Then I tried this and that and that and this and the best I could come up with is represented by the following:
proof 01-19 3.PNG

I can't seem to get beyond this....has anyone got any ideas? I have tried things like writing 5 as (18-13) etc., but had no luck.
 
allegansveritatem said:
Here is what I'm being asked to prove for every positive integer n:
View attachment 24513

First I tested with n=1:
View attachment 24514

Then I tried this and that and that and this and the best I could come up with is represented by the following:
View attachment 24515

I can't seem to get beyond this....has anyone got any ideas? I have tried things like writing 5 as (18-13) etc., but had no luck.
Click to expand...
There are 3 steps in induction proofs. You posted the first one. The second is just as easy as the first. Please post it. Then we'll work on the 3rd step - the actual proof.
 
allegansveritatem said:
Here is what I'm being asked to prove for every positive integer n:
View attachment 24513

First I tested with n=1:
View attachment 24514

Then I tried this and that and that and this and the best I could come up with is represented by the following:
View attachment 24515

I can't seem to get beyond this....has anyone got any ideas? I have tried things like writing 5 as (18-13) etc., but had no luck.
Click to expand...
Change the first 10 into (9+1) and use the distributive law.
 
In mathematics you sometimes need to make things work. As Harry_the-cat pointed out, change 10 into 9+1. Then the multiple of 10 becomes a multiple of 9 plus a multiple of 1. Clearly 9 goes into the multiple of 9. Does 9 go into the multiple of 1????
 
Maybe (probably not but you need to try things) factor out a 10^n from (10^(n+1) + 3*10^n)
 
lev888 said:
There are 3 steps in induction proofs. You posted the first one. The second is just as easy as the first. Please post it. Then we'll work on the 3rd step - the actual proof.
Click to expand...
Well, I thought that the second step--which as I understand it, is just a formality--I took as understood. I will go back to this problem today and come back with a fuller presentation.
 
Jomo said:
In mathematics you sometimes need to make things work. As Harry_the-cat pointed out, change 10 into 9+1. Then the multiple of 10 becomes a multiple of 9 plus a multiple of 1. Clearly 9 goes into the multiple of 9. Does 9 go into the multiple of 1????
Click to expand...
Exactly. Nine will go nicely into nine but it will have a hard time fitting into that one.
 
Jomo said:
Maybe (probably not but you need to try things) factor out a 10^n from (10^(n+1) + 3*10^n)
Click to expand...
I think I have tried this already in one of my early attempts but I will keep hacking at it and see what I can come up with. This purely symbolic math is lean, mean stuff.
 
allegansveritatem said:
I can see the why of the 9 but not of the 1 here.
Click to expand...
There are two ways you might think of this, which work together nicely. One is that you want to use the assumption that 10n+1+3⋅10n+510^{n+1}+3\cdot 10^n+5 is a multiple of 9, so you'd like to have that expression present in your new expression. The other is that you want 9's wherever you can.

But you don't always know ahead of time what will work, so you have to just try things. Have you tried following the suggestion to see what will happen?? 10n+2+3⋅10n+1+5=10(10n+1+3⋅10n)+5=(9+1)(10n+1+3⋅10n)+510^{n+2}+3\cdot 10^{n+1}+5 = 10(10^{n+1}+3\cdot 10^n)+5= (9+1)(10^{n+1}+3\cdot 10^n)+5. Now distribute this: 9(10n+1+3⋅10n)+1(10n+1+3⋅10n)+59(10^{n+1}+3\cdot 10^n)+1(10^{n+1}+3\cdot 10^n)+5. What can you do next?
 
Jomo, why surprised face? I thought that's just the standard way to do these division type of induction problems.
 
Everyone is asking you to use your assumption but as lev888 pointed out you did not write it down. Might that be why you are having so much trouble? If something is there on the paper you might use it. If it is just something in your head somewhere you might not use it. Write down everything that you know for the problem and try to use it!
 
Jomo said:
Everyone is asking you to use your assumption but as lev888 pointed out you did not write it down. Might that be why you are having so much trouble? If something is there on the paper you might use it. If it is just something in your head somewhere you might not use it. Right down everything that you know for the problem and try to use it!
Click to expand...
"Right" down everything.... good one Jomo!
 
Harry_the_cat said:
Jomo, why surprised face? I thought that's just the standard way to do these division type of induction problems.
Click to expand...
The surprised face is because I am amazed that the student can't see this. I guess you can lead a horse to water but can't make the horse doing it.
 
Harry_the_cat said:
"Right" down everything.... good one Jomo!
Click to expand...
Its been a long year. What can I say? I need to take a break from the forum but I was told that if I do that I will not get my 15% raise in pay so I have to stick around. Is it just January?!
 
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