![summation - Confusion about how to prove $\sum_{i=0}^n 2^i = 2^{n+1}-1$ for all $n\geq 0$ by induction - Mathematics Stack Exchange summation - Confusion about how to prove $\sum_{i=0}^n 2^i = 2^{n+1}-1$ for all $n\geq 0$ by induction - Mathematics Stack Exchange](https://i.stack.imgur.com/NauCL.jpg)
summation - Confusion about how to prove $\sum_{i=0}^n 2^i = 2^{n+1}-1$ for all $n\geq 0$ by induction - Mathematics Stack Exchange
![summation - Is there any. elementary formula for the sequence$\sum_{k=1}^{n }\left(2k-1\right)\left(\frac{1}{2}\right)^{k}$ - Mathematics Stack Exchange summation - Is there any. elementary formula for the sequence$\sum_{k=1}^{n }\left(2k-1\right)\left(\frac{1}{2}\right)^{k}$ - Mathematics Stack Exchange](https://i.stack.imgur.com/mMJNf.png)
summation - Is there any. elementary formula for the sequence$\sum_{k=1}^{n }\left(2k-1\right)\left(\frac{1}{2}\right)^{k}$ - Mathematics Stack Exchange
![proof writing - Prove for all n∈N $1^2+3^2+5^2+...+(2n-1)^2=\frac{4n^3-n}{3}$ - Mathematics Stack Exchange proof writing - Prove for all n∈N $1^2+3^2+5^2+...+(2n-1)^2=\frac{4n^3-n}{3}$ - Mathematics Stack Exchange](https://i.stack.imgur.com/gkQWr.png)