query: Find a formula for $1 + 3 + 5 + ... +(2n - 1)$, for $n \ge 1$, and prove that your formula is correct.
positive_passages.docid: 17491
positive_passages.text: Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$
negative_passages.docid: 14306
negative_passages.text: What is the formula for $\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{n(n+1)}$
negative_passages.docid: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
negative_passages.docid: 61871
negative_passages.text: Formula for the $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5, ...$
negative_passages.docid: 110441
negative_passages.text: Closed form for a formula with a summation over $i\binom{n-i}{k-1}$, and combinatorial proof?
negative_passages.docid: 93572
negative_passages.text: How to determine if $n (n \geq 2)$ given numbers can be summed up to a certain number?
negative_passages.docid: 42831
negative_passages.text: How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$
negative_passages.docid: 50599
negative_passages.text: Prove that $\frac{1}{1*3}+\frac{1}{3*5}+\frac{1}{5*7}+...+\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}$
negative_passages.docid: 104753
negative_passages.text: $\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?
negative_passages.docid: 96268
negative_passages.text: Prove that $5$ divides $3^{3n+1}+2^{n+1}$
negative_passages.docid: 131400
negative_passages.text: Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$
negative_passages.docid: 107098
negative_passages.text: Find an explicit formula for the recurrence $a_n = n(a_{n-1} + a_{n-2})$ knowing that $ a_0=1$ and $a_1=2$.
negative_passages.docid: 11407
negative_passages.text: Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$
negative_passages.docid: 94389
negative_passages.text: Proving formula for product of first n odd numbers
negative_passages.docid: 19909
negative_passages.text: if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$
negative_passages.docid: 86900
negative_passages.text: Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
negative_passages.docid: 131771
negative_passages.text: Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$