Induction proof n<10^n- stuck

Natasha13100

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I am trying to prove n<10^n. Can I get some help?
So far:
for n=1, n<10^n because 1<10.
assume for n=k, k<10^k so n<10^n
therefore, k+1<10^k+1 (if a<b, a+c<b+c for any c)
this is where I'm stuck. Here is my guess:
for any positive integer k, 10^k will be a positive integer.
therefore, 10^(k+1)=10(10^k)>(10^k)+1.
thus, 10^(k+1)>k+1 (if a<b and b <c, a<c)
Only the last part has been proven so far so I don't think I'm doing this correctly.
 
You are very close to being there, looks like just a little rewording to me:

As you pointed out 10n+1 = 10 10n, so let's write that as
10n+1 = 10 10n = (1 + 9) 10n = 10n + 9 10n > n + 9 10n
since 10n > n. Now, 10n > 1 for n>1 since 10k > k > 1, k = 2, 3, 4, ..., n. Can you finish it from there?
 
Remember

I am trying to prove n<10^n. Can I get some help?
So far:
for n=1, n<10^n because 1<10.
assume for n=k, k<10^k so n<10^n
therefore, k+1<10^k+1 (if a<b, a+c<b+c for any c)
this is where I'm stuck. Here is my guess:
for any positive integer k, 10^k will be a positive integer.
therefore, 10^(k+1)=10(10^k)>(10^k)+1.
thus, 10^(k+1)>k+1 (if a<b and b <c, a<c)
Only the last part has been proven so far so I don't think I'm doing this correctly.

I haven't used math induction in a while but remember that 10^(k+1) = 10^(k) + 10^(1).
See if that helps...
 
I haven't used math induction in a while but remember that 10^(k+1) = 10^(k) + 10^(1).
See if that helps...

No, 10^(k + 1) = (10^k)(10^1), or if you like to use the asterisk also, (10^k)*(10^1).
 
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You are very close to being there, looks like just a little rewording to me:

As you pointed out 10n+1 = 10 10n, so let's write that as
10n+1 = 10 10n = (1 + 9) 10n = 10n + 9 10n > n + 9 10n
since 10n > n. Now, 10n > 1 for n>1 since 10k > k > 1, k = 2, 3, 4, ..., n. Can you finish it from there?

Thank you so much. That was just the help I needed. (I'm assuming you used spaces instead of multiplication signs.)
 
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