The problem that I have is: Prove by induction that 21 divides 4n+1 + 52n-1 So far I have: Base Case: $n = 1$ $4^{1+1} + 5^{2-1} = 4^2 + 5^1 = 16 + 5 = 21...

So I got into wolfram to see whether I solved correctly an exercise. More specifically the first-order derivative of $y(x)=e^x (ax^4+bx^3+cx^2+dx+q)$. Im ...

Find the global maximum and global minimum of $f : \mathbb{R^2} \rightarrow \mathbb{R}$ given by $f(x,y) = x^2 + (y-1)^2$ on the set $E = \{(x,y) \in \mat...

The mean of the binomial distribution, i.e. the expectation for number of events, is $np$. I've seen this proven by rearranging terms so that $np$ co...

How this equality is verified. $$\arctan(2\sqrt2)+2\arctan(\sqrt2)=\pi$$

For example, in $\displaystyle{\frac{a+bi}{n+zi}}$, you would multiply both by the complex conjugate of the denominator, $n-zi$, to get rid of the complex...

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He to...

I cant seem to solve this problem. A train leaves point A at 5 am and reaches point B at 9 am. Another train leaves point B at 7 am and reaches point A at...

I'm trying to find the antiderivative of $A(x)=\cos(x)\log(1+\cos(x))$ By using integration by parts I get : $$\int A(x)\, dx = \sin(x)\ln(1+\cos(x))...

I don't know what method to approach to get the required proof.