Can anyone help me confirm if I have solved this correctly?
Many thanks.
Q. 3^{n} > n^{2} for n > 2, n .
Attempt: Step 1: For n = 2,
3^{2} = 9 & 2^{2 }= 4
Since 9 > 4, the statement is true for n = 2.
Step 2: Assume the statement is true for n = k, i.e. assume 3^{k} > k^{2}.
We must now show that the statement is true for n = k + 1,
i.e. 3^{k+1 }> (k + 1)^{2}
3^{k+1} = 3^{1} 3^{k}...(3^{k} > k^{2}...assumed)
If it can be shown that 3(k^{2}) > (k + 1)^{2}, then 3^{k+1} > (k + 1)^{2}
3(k^{2}) > (k + 1)^{2 }if 3k^{2} - (k + 1)^{2} > 0
if 3k^{2} - k^{2} - 2k - 1 > 0
if 2k^{2} - 2k - 1 > 0
if 2k^{2} - 2k + 1 - 1 - 1 > 0...completing the square
if 2k^{2} - 2k + 1 - 2 > 0
if (2k - 1)(k - 1) - 2 > 0...true for k > 2 = 3^{k+1} > (k + 1)^{2}
Therefore, the statement is true for n = k + 1, if true for n = k. Thus the statement is true for all n > 2, n .
I'm posting an example from the book to demonstrate my approach. The author uses the values specified in the question to completion, hence my sticking with the 2 throughout my attempt. The book also introduces 'completion of the square' but only uses simplistic examples, which I have now had to read up on independently of the book, in lieu of better examples. This entire chapter has been a complete mess, compared to the others!!
I think the problem in post #1 starts with n = 2 because the proof method of the inductive step shown in a similar problem in post #7's attachment works only for n ≥ 2. The inductive step shows that implies . The conclusion is true for k = 1; however, if we replace by in the conclusion's left-hand side, then the resulting inequality is true only for k ≥ 2. This follows from the fact that the positive root of 2kČ - 2k - 1 = 0 is . Of course, it it possible to break the proof of the inductive step into two cases: k = 1 and k ≥ 2.
Speaking about quadratic equations and inequalities, I personally never complete the square or guess the roots based on Vieta's formulas; I always use the quadratic formula.
I think what you've done is okay (although as some pointed out, you need to be more careful with your notation), except for one thing, which is that you did not complete the square correctly. You used the method for when the coefficient of k^2 is 1 ... in this case it is 2. You'd end up with roots, which I'm not sure would be obvious. You could try this also:
From 2k^2 - 2k - 1 > 0
2k^2 - 2k > 1
2k ( k - 1 ) > 1 is true for k > 1