query: Why: $2^{n+1} = 1 + \sum_{i=0}^{n}2^i$
positive_passages.docid: 26607
positive_passages.text: The idea behind the sum of powers of 2
negative_passages.docid: 101017
negative_passages.text: What is the sum of $\sum\limits_{i=1}^{n}ip^i$?
negative_passages.docid: 69131
negative_passages.text: Different proofs of $\,a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?
negative_passages.docid: 146533
negative_passages.text: Having problems proving $ \sum_{r=0}^{n}(r+1) \binom{n}{r} = (n+2)2^{n-1} $
negative_passages.docid: 151646
negative_passages.text: Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
negative_passages.docid: 98808
negative_passages.text: Finite Sum $\sum_{i=1}^n\frac i {2^i}$
negative_passages.docid: 26446
negative_passages.text: Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$
negative_passages.docid: 60860
negative_passages.text: Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
negative_passages.docid: 110996
negative_passages.text: Sum of the form $r+r^2+r^4+\dots+r^{2^k} = \sum_{i=1}^k r^{2^i}$
negative_passages.docid: 133905
negative_passages.text: Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
negative_passages.docid: 14064
negative_passages.text: Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$
negative_passages.docid: 155014
negative_passages.text: How to justify the definition of summation $s_n=\sum_{i=1}^n a_i=a_1+\cdots+a_n$?
negative_passages.docid: 120976
negative_passages.text: Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
negative_passages.docid: 147165
negative_passages.text: Does the fact that $\sum_{n=1}^\infty 1/2^n$ converges to $1$ mean that it equals $1$?
negative_passages.docid: 22129
negative_passages.text: Why is $\sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|$ true?
negative_passages.docid: 52206
negative_passages.text: How to prove with double counting technique that $1+2+\dots+2^n=2^{n+1} -1$?
negative_passages.docid: 58428
negative_passages.text: Proving $\sum_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1\choose a+b+1}$