I have these 2 questions from my discrete Math assignment and i can't figure out how to deal with it. Inexperienced friends of mine say that the question is wrong. Can anyone help me solve it?
hehe, your questions are wrong. the first is not an inequality, it should read $\displaystyle 2n + 1 ~{\color{red} < }~2^n$ for $\displaystyle n = 3,4, \dots$. the second question has no real solutions, much less integer solutions.
for the first, here are the steps:
Let $\displaystyle P(n):~2n + 1 < 2^n$ for $\displaystyle n = 3,4, \dots$
Verify that $\displaystyle P(3)$ holds. this is your base case.
Now assume $\displaystyle P(n)$ holds, and show that $\displaystyle P(n + 1)$ holds.
that is, assume $\displaystyle 2n + 1 < 2^n$ and show that it implies $\displaystyle 2(n + 1) + 1 < 2^{n + 1}$
good luck!
For the second one, I think that the parentheses are meant to be "integer part" symbols:
$\displaystyle \left\lfloor\frac{\lambda^2}4\right\rfloor = \left\lfloor\frac{\lambda-1}2\right\rfloor\, \left\lfloor\frac{\lambda+1}2\right\rfloor$
This is true when λ is an odd integer. In fact, if λ = 2k+1 then both sides are equal to k(k+1).