proof of mathematical induction?

Question

how do I prove 60 * 5^n < 6^n for all n >= some n1?

This inequality holds true when n >22 but how do i prove this by mathematical induction?

Answer 1

Suppose it is true for some n . I do not know because you say 22 I can compute and check

So 1st step is right

let it be true for some n

we need to prove for n+1

60*5^n < 6^n
multiply both sides by 5
60*5^(n+1) < 6^n . 5 ...1

but 6^n. 5 < 6^n . 6 as 6^n > 0
so 6^n. 5 < 6^(n+1) ....2

from 1 and 2
60.5^(n+1) < 6^(n+1)


QED

Answer 2

If 66 * 5^n < 6^n is true for some n, then multiplying both sides by 6 gets you 6/5 * 66 * 5^(n+1) < 6^(n+1). We know that 66 * 5^(n+1) < 6/5 * 66 * 5^(n+1). Therefore 66 * 5^(n+1) < 6^(n+1). If this is true for n+1, then it's also true for n+2, n+3, etc., and thus, this inequality is proven by mathematical induction.

Answer 3

Consider the function f(n)=(6/5)^n. The limit as n goes to infinity is infinity (since 6/5>1), so there is some n_1 such that (6/5)^(n_1) > 60. Moreover, the function f is increasing. For this, let the variable have real values and take a derivative: f ' (x) = (6/5)^x * ln(6/5) > 0. Hence, for n > n_1, 60 < (6/5)^(n_1) < (6/5)^(n_1). Induction is not necessary, but you could artificially put it in if you want to.

Answer 4

First, prove that it is true for n = 23

60*5^23 < 6^23

715255737304687500 < 789730223053602816

Next we ASSUME it is true for some n > 22, that is
60 * 5^n < 6^n

We want to show that assuming it is true for n, it must be true for n+1

60 * 5^n * 5 < 6^n * 5 and 6^n * 5 < 6^n * 6

Therefore 60*5^(n+1) < 6^(n+1) as required

Since we know its true for 23, we know its true for 24
Since we know its true for 24, we know its true for 25
...

So we know it is true for ALL n > 22 by induction

Answer 5

We can prove it by taking logs
lg60+nlg5 n[lg6-lg5]>lg60
n>lg60/lg1.2
>1.77815/.079181
>22.4
If we substitute n+1 for n
we get a similar results.

Answer 6

log60+nlog5 log60+nlog-nlog6<0
nlog6-nlog5-log60>0
nlog(6/5)>log60
0.0792n>1.7782
n>1.7782/0.0792
n>22.45