Skip to main content
3 votes

Trying to prove that if $\frac{f(x+1)}{f(x)} = b,$ where $b$ is constant, then $f(x)$ is exponential.

As you have shown, for $k\in\mathbb{Z}$ you have $$ f(x+k)=b^kf(x) $$ for all $x\in\mathbb{R}$. However, this is not enough to prove that $f(x)$ is exponential. Instead: Lemma: $\frac{f(x+1)}{f(x)} = ...
Dark Malthorp's user avatar
2 votes
Accepted

Prove that $\log_2(n) \le \sqrt{2n}$ for all $n\ge1$

The question chose $k$ to be the smallest integer such that $n\le 2^k$, so $n \not \le 2^{k-1}$ but $2^{k-1} < n$. Multiply this by $2$ to get $2^k < 2n$: $$2^{k-1} < n \le 2^k < 2n$$ For $...
peterwhy's user avatar
  • 22.5k
1 vote

How to more formally prove this inequality

You can simplify the reasoning, getting rid of the absolute values, by noting that $|y-n|=y-n$ and $|y-(n+1)|=n+1-y.$ Then you only need to prove that $y-n>n+1-y$ which is equivalent to $2y > 2n+...
Lieven's user avatar
  • 729
1 vote

Prove that the determinant of a matrix with a+b on diagonals and a on off diagonals is (b^(n-1))(na+b)

Consider A the matrix for which every coefficient is equal to $a$. Then the determinant your are looking for is $$\Delta := \det (A+bI) = (-1)^nP_A(-b)$$ if we note $P_A$ the characteristic polynomial ...
Timothe Schmidt's user avatar
1 vote
Accepted

Prove that the determinant of a matrix with a+b on diagonals and a on off diagonals is (b^(n-1))(na+b)

Subtract the last column from all other columns. This operation preserves the determinant. The result is all $b$'s on the diagonal except in the lower right corner where the original $a+b$ entry ...
Lieven's user avatar
  • 729
1 vote

Assuming the conclusion while trying to prove an inequality

There is the well known result for positive numbers $\frac ab<\frac cd\implies \frac ab<\frac{a+c}{b+d}<\frac cd$ We have trivially $\frac{2n}{n+1}<2$ and since $\ln(\cdot)$ is an ...
zwim's user avatar
  • 28.7k
1 vote

Assuming the conclusion while trying to prove an inequality

You can build a valid argument without assuming the conclusion, by reversing the steps in your original deduction. Each of the steps has an equivalence arrow pointing both ways, so reversing the steps ...
Lieven's user avatar
  • 729
1 vote

Trying to prove that if $\frac{f(x+1)}{f(x)} = b,$ where $b$ is constant, then $f(x)$ is exponential.

actually, take $f(0) = 1$ and $f(1) = b,$ but fill in values of $f(x) $ for $0 < x < 1$ any way you like for continuous piecewise linear fill in a line segment, $f(x) = (b-1)x + 1$ for $...
Will Jagy's user avatar
  • 140k
1 vote

Mathematical Induction: Strong vs Weak Form

I would say it is because weak induction really has that 'domino stone' feel to it, which for novices really helps to grasp induction and why it works. And also: for many induction problems weak ...
Bram28's user avatar
  • 101k

Only top scored, non community-wiki answers of a minimum length are eligible