Good night, fardeen gen. (Or good afternoon, evening, morning, I'm not sure what time of day it is in India)
First off, I believe that you have copied the equation wrong. It should be
Test at values of you should see that you need to flip the sign (unless I'm wrong)
Onto the solution
Cross-multiply and then remove logs. The equation becomes
When we test , we see that equality holds.
Test and and we see that the inequality holds. Assuming that this pattern will hold true for all natural numbers greater than
We now wish to prove
So we have
With a little algebraic manipulation we shall now prove
Assume that this is not true, i.e.
Because the coefficient of is positive, the curve has a max. turning point and because of our assumption it means that the entire curve is below the x-axis, i.e. it has no real roots.
For this equation to have no real roots the discriminant
To determine the values for n under which our assumption holds, we use the quadratic formula
This value of is complex, however the question asks us to prove this statement of natural numbers.
Because our assumption holds true only when n is complex, we must reject our assumption due to question constrains.
Therefore, the statement must hold true for all natural numbers greater than .
Equality holds at .
Hence, our original statement
is true for all natural numbers.
End of solution
I'm not sure how valid, so I (and fardeen_gen) would appreciate it if a more experienced forum member critique this solution
Also, I would appreciate it if someone could critique my method of proof writing and give me tips on how to better my proof-writing skills.
Merci beaucoup in advance.
Hope this was useful