Induction proof n<10^n- stuck

[Natasha13100]

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.

 

[Ishuda]

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 10^k > k > 1, k = 2, 3, 4, ..., n. Can you finish it from there?

 

[bradgregoryatm]

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...

 

[lookagain]

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).

 

[Natasha13100]

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 10^k > 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.)

 

[Ishuda]

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

You are very welcome.