1)
[tex]\it \dfrac{1}{(7n-6)(7n-1)}=\dfrac{1}{7}\Big(\dfrac{1}{7n-6}-\dfrac{1}{7n+1}\Big)\\ \\ \\ S=\dfrac{1}{7}\Big(1-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{15}+\dfrac{1}{15}-\dfrac{1}{2}+\dfrac{1}{7n-6}-\dfrac{1}{7n+1}\Big)=\\ \\ \\ =\dfrac{1}{7}\Big(1-\dfrac{1}{7n+1}\Big)=\dfrac{1}{7}\cdot\dfrac{7n}{7n+1}=\dfrac{n}{7n+1}[/tex]
2)
[tex]\it a_1=1=3\cdot2^1-5\\ \\ a_2=7=3\cdot2^2-5\\ \\ a_3=19=3\cdot2^3-5\\.\\.\\.\\.\\a_{20}=3\cdot2^{20}-5\\ \\ \\ S=3(2+2^2+2^3+\ ...\ +2^{20})-5\cdot20=3\cdot2\cdot\dfrac{2^{20}-1}{2-1}=6(2^{20}-1)[/tex]
