1. ✅

$$ 2+2^2+2^3+...+2^n=2^{n+1}-2\space \space \forall n\in \N\newline \sum^n_{i=1}2^i=2^{n+1}-2 $$

• Soluzione

1. ✅

$$ 1^3+2^3+3^3+...+n^3=(1+2+...+n)^2\space \space\space \forall n\in \N\newline \sum^{n}_{i=1}i^3=\bigg(\frac{n(n+1)}{2}\bigg)^2 $$

• Soluzione

1. ✅

$$ 2^{n-1}\le n!\space \space \forall n\in \N $$

• Soluzione

1. ✅

$$ \sum^n_{k=1}\frac{4k^2+2k-1}{(2k+1)!}=1-\frac{1}{(2n+1)!}\space \space \forall n\in \N $$

• Soluzione

1. ✅

$$ \sum^n_{K=1}\frac{4k^2+6k+1}{(2k+2)!}=\frac{1}{2}-\frac{1}{(2n+2)!}\space \space \forall n\in \N $$

• Soluzione