(Solved):   1. Prove the following Theorems, by using Mathematical Induction: a) \( a+a r+a r^{2}+\cdo ...

 

1. Prove the following Theorems, by using Mathematical Induction: a) \( a+a r+a r^{2}+\cdots+a r^{n}=\frac{a\left(r^{n+1-1}\right)}{r-1} \) for all real numbers \( a, r \) with \( r \neq 1 \) b) \( 1(1 !)+2(2 !)+3(3 !)+\cdots+n(n !)=(n+1) !-1 \) c) \( 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(2 n+1)(n+1)}{6} \) for all \( n \in \mathbb{N} \). d) Prove that 3 is a factor of \( 4^{n}-1 \) for all positive integers. e) \( \left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\cdots+\left(\begin{array}{l}n \\ n\end{array}\right)=2^{n} \), for every positive integer \( n \). f) Prove that \( n<2^{n} \) for every positive integer \( n \).