# Tagged Questions

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### Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ Using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial ...
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### Prove $\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) [duplicate]

Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$\sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$$ holds?
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### Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction

I am just starting out learning mathematical induction and I got this homework question to prove with induction but I am not managing. $$\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$$ ...
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### Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
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### Finite Sum of Power?

Sorry for the remedial math but: Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^x$$ For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?
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### Proving $\sum\limits_{i=0}^n i 2^{i-1} = (n+1) 2^n - 1$ by induction

I'm trying to apply an induction proof to show that $((n+1) 2^n - 1$ is the sum of $(i 2^{i-1})$ from $0$ to $n$. the base case: L.H.S = R.H.S we assume that $(k+1) 2^k - 1$ is true. we need to ...
### Proving $\sum\limits_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$
Show that $$\sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$$ So for odd $n$ we have an even number of terms. So $\binom{n}{k} = \binom{n}{n-k}$ which have opposite signs. Thus the sum is $0$. For even $n$ ...