I've been stuck on this question for a good while now, and I think I need some pointers.  $(1+x)^n = nC_0 + nC_1x + nC_2x^2 ... + nC_{n-1}x^{n-1} + nCnx^n$ where n greater than or equal to 1  I'm at the point where I replace n with k+1 and solve, but no matter what I do I cannot get the sides of the equation to look the same.
123923
I'm trying to prove binomial theorem by induction, but I'm a little stuck. I would look at online resources as this problem has been done many times, but the version I am trying to prove the binomial theorem in a different form.  $$(1 + x)^n = \sum_{k = 0}^{n} \binom{n}{k} x^k$$  I'm mostly confused as to how I can make the left side be equivalent to a summation, any help is appreciated. Try to hint me along!
7879
I'm trying to solve this equation:  $$2-x=-\sqrt{x}$$ Multiply by $(-1)$: $$\sqrt{x}=x-2$$ power of $2$: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$    But $x=1$ is not a correct solution to the original equation.  Why have I got it? I've never got a wrong solution to an equation before. What is so special here?
66019
Find polynomials $A(x)$ and $B(x)$ such that $A(x)P(x) + B(x)Q(x) = x + 1$ for all $x$ where $P(x) = x^4 - 1$ and $Q(x) = x^3 + x^2$. I'm stumped on this question. I know that I'm supposed to apply the extended version of Euclid's algorithm for polynomials but I'm unsure of how to do that. I thought about trying to create some kind of linear system but guessing arbitrary coefficients but that wouldn't work as $A$ and $B$ don't have fixed degrees.
114196
Let $x_n$ be the unique non-negative real solution of a equation $x^n+x^{n-1}+x-1=0$. Prove the sequence {$x_n$} is increasing and converges to 1.  I found out that $ 0 &lt; x_n &lt; 1$ but just that..
73413
I want to prove the following equality:  $$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$$  So I had an idea to use $((1+x)^n)'=n(1+x)^{n-1}$  So I could just use the binomial theorem and let $$x=2 \implies (n3^{n-1})$$ and then modify the sum into that one on the left side.  So I need to prove that : $$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n(1+x)^{n-1}$$ if$$x=2$$ Any help would be appreciated.
113190
$x_{n+2} = x_{n+1} + 20x_n + n^2 + 5^n \text{ with } x_0 = 0 \text{ and } x_1 = 0$  How would you solve this recursive relation? I have the homogenous solution, but am having issues with the in-homogenous one.
60293
I want to know if there is another way to solve the equation:  $$(x+1)^y-x^z=1$$  for positive integers $x,y,z$ greater than $1$ without using the Catalan identity. Does exist? By now I only know that $x\mid y$.
105975
Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n &lt; 0$ I have to prove that the Polynomial $P(x)$ has at least one positive root how can I prove it? Any ideas?
63201
Find all real solutions to the equation:$$4^x+6^{x^2}=5^x+5^{x^2}$$  Obviously $0$ and $1$ satisfy this equation, but here $x$ is a real number, and I do not know how to do the next step, in fact, I guess this equation is not in addition to $0$ and $1$ other than the root. But you know, I need a proof! Could anyone help me? Thanks a lot!
3610
Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$  I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? Thank you.
82351
Prove that for all $n\in\mathbb N$, $9 \mathrel| (4^n+6n-1)$.  I have shown the base case if $n=0$ (which is a natural number for my course).  I'm now trying to prove that $9k=4^n+6n-1$.  I substituted $n+1$ for $n$, and have $4^{(n+1)}+6(n+1)-1$, which equals $4*4^n+6n+5$.  I'm stuck. Is that not the right way to go? I don't know how to get from the “$+5$” to “$-1$” so I can substitute $9k$ to the right side of the equation. Any help would be greatly appreciated.
108956
I have literally no idea how to do this. I tried writing out $f(x+1)$ and rewriting it but didn't seem to get anywhere. I suspect I should use the degree of the polynomials but don't know how.
33410
I have to prove that: $$ \binom n 0 ^2  + \binom n 1 ^2  + \dots + \binom n n ^2 = \binom { 2n} n $$  I don't want a complete solution, but only a hint.
91139
How many solutions are there to the equation      $x_1 + x_2 + x_3 + x_4 + x_5 = 21$    where $x_1, i = 1,2,3,4,5$ is a nonnegative integer such that $0 ≤ x_1 ≤ 3, 1 ≤ x_2 &lt; 4$, and $x_3 ≥ 15$?  I have correctly completed the previous parts of the question but am having trouble with this particular restriction.  The total number of solutions to the equation is $C(25,21) = 12650$.  I have tried to do each restriction individually:  For $0 ≤ x_1 ≤ 3$, I solved this by subtracting the number of solutions when $x_1 ≥ 4$ from the total number of solutions. This is $C(25,21) - C(21,17) = 5985$.  For $1 ≤ x_2 &lt; 4$, I have $C(24,20) - C(21,17) = 5985$.  For $x_3 ≥ 15$, I have $C(10,6) = 210$.  However, the restriction is when these are all together. I am not sure how to proceed. The answer is $106$.
5287
Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$  a) Show that $x_{n} &lt; x_{n+1}$  b) Show that  $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$  Hint : Square $x_{n+1}$ and factor a 2 out of the square root  c) Hence Show that  $x_{n}$ is bounded above by 2. Deduce that $\lim\limits_{n\to \infty} x_{n}$ exists.  Any help? I don't know where to start.
51742
I need to find the values of $a$ and $n$ in the following  $$ (1 + ax)^n  = \sum_{i=0}^\infty \binom{2i}{i}x^i$$  How can I compare things and show that $a=-4$ and $n = -{1 \over 2}$. It's easy to expand $1 \over \sqrt{1 -4x}$ see that it's valid but I'm looking other way around.
98209
The equation $X^{n}  + Y^{n}  = Z^{n}$    , where $n   \geq  3$     is a natural number, has no solutions at all where $X,Y,Z$     are integers.   My solution:  False. Because if we let $X=0  ,Y=0$,  then $0^{3}+0^{3}=0^{3}$  . Thus there is a solution for the equation $X^{n}  + Y^{n}  = Z^{n}$,   where $X,Y,Z$   are integers.  Can anyone please give feedback on my answer and state whether it is correct on not.