query: Demonstration of sum of powers of $2$
positive_passages.docid: 26790
positive_passages.text: Proving the geometric sum formula by induction
negative_passages.docid: 41789
negative_passages.text: Fermat's theorem on sums of two squares
negative_passages.docid: 111098
negative_passages.text: Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
negative_passages.docid: 31977
negative_passages.text: How to prove sum of powers property of roots of unity?
negative_passages.docid: 142132
negative_passages.text: How do you prove $\sum \frac {n}{2^n} = 2$?
negative_passages.docid: 122149
negative_passages.text: How to prove that $\sum_{i<j}(X_i-X_j)^2=n\sum_{i=1}^{n}(X_i-\bar{X})^2$
negative_passages.docid: 97984
negative_passages.text: Express as sum of powers
negative_passages.docid: 98440
negative_passages.text: Show that any non-negative rational integer may be written as the sum of two squares if it is of the form....
negative_passages.docid: 4201
negative_passages.text: Prove $\sum\binom{n}{k}2^k = 3^n$ using the binomial theorem
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 118372
negative_passages.text: Combination proof for $n(n+1)2^{n-2}=\sum_{k=1}^{n}k^2\binom{n}{k}$
negative_passages.docid: 52115
negative_passages.text: Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction
negative_passages.docid: 158863
negative_passages.text: How to prove $\sum_p p^{-2} < \frac{1}{2}$?
negative_passages.docid: 11612
negative_passages.text: How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
negative_passages.docid: 84213
negative_passages.text: How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
negative_passages.docid: 145214
negative_passages.text: Showing the power set is equinumerous to ${}^X2$
negative_passages.docid: 109599
negative_passages.text: A finite sum involving the binomial coefficients and the harmonic numbers