query: How to prove that $S_n = n^{2}$?
positive_passages.docid: 142931
positive_passages.text: Problem on Euler's Phi function
negative_passages.docid: 126312
negative_passages.text: How can you prove $\binom {2n}{2} = 2\binom{n}{2} + n^2$?
negative_passages.docid: 44391
negative_passages.text: How can I prove $n - 2\sqrt{n} = \Theta(n)$
negative_passages.docid: 88124
negative_passages.text: $M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$
negative_passages.docid: 52157
negative_passages.text: Prove that $d(n)\leq 2\sqrt{n}$
negative_passages.docid: 99052
negative_passages.text: Prove that $( 1 + n^{-2}) ^n \to 1$.
negative_passages.docid: 64782
negative_passages.text: How to prove if $n$ is prime and $n | a^2$ then $n | a$?
negative_passages.docid: 88144
negative_passages.text: Induction Proof $n! < n^n$
negative_passages.docid: 117681
negative_passages.text: For which Natural $n\ge2: \phi(n)=n/2$
negative_passages.docid: 78190
negative_passages.text: How to prove that for all natural numbers, $4^n > n^3$?
negative_passages.docid: 54369
negative_passages.text: Proof of $n^2 \leq 2^n$.
negative_passages.docid: 27580
negative_passages.text: Use mathematical induction to prove $ (2n)!\geq 2^n(n!)^2$ for $n \in \mathbb{N}$
negative_passages.docid: 37906
negative_passages.text: Prove $s_n$ is a Cauchy sequence for $|s_{n+1}-s_n|\lt2^{-n}$
negative_passages.docid: 2455
negative_passages.text: Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
negative_passages.docid: 6165
negative_passages.text: Proof by Induction $n^2+n$ is even
negative_passages.docid: 10604
negative_passages.text: Prove $3^n > n^2$ by induction
negative_passages.docid: 15838
negative_passages.text: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$