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1) $a_1=2;d=5;S_n=156;\\\\S_n=\frac{2a_1+(n-1)*d}{2}*n;\\\\\frac{2*2+5*(n-1)}{2}*n=156;\\\\(4+5n-5)n=312;\\\\5n^2-n-312=0;\\\\D=(-1)^2-4*5*(-312)=6241=79^2;\\\\n_1=\frac{1-79}{2*5}<0;\\\\n_2=\frac{1+79}{2*5}=8;\\\\n=8$

2) $b_4=20;b_5=30;\\\\q=\frac{b_5}{b_4}=\frac{30}{20}=1.5$

3) $b_5-b_1=160;b_4-b_2=48;\\\\b_1q^4-b_1=160;b_1q^3-b_1q=48;\\\\b_1(q^4-1)=160;b_1(q^3-q)=48;\\\\\frac{q^4-1}{q^3-q}=\frac{160}{48};\\\\\frac{(q^2-1)(q^2+1)}{q(q^2-1)}=\frac{10}{3};\\\\3(q^2+1)=10q;\\\\3q^2-10q+3=0;\\\\D=(-10)^2-4*3*3=64=8^2;\\\\q_1=\frac{10-8}{2*3}=\frac{1}{3};\\\\q_2=\frac{10+8}{2*3}=3;\\\\b_1(1)=\frac{160}{q^4-1}=\frac{160}{(\frac{1}{3})^4-1}=-162;\\\\b_1(2)=\frac{160}{q^4-1}=\frac{160}{3^4-1}=2$

4) $b_1=\frac{2}{3};b_2=\frac{4}{9};\\\\q=\frac{b_2}{b_1}=\frac{\frac{4}{9}}{\frac{2}{3}}=\frac{2}{3};|q|<1;\\\\S=\frac{b_1}{1-q}=\frac{\frac{2}{3}}{1-\frac{2}{3}}=2$

5) $b_1=\frac{2}{3};b_2=\frac{1}{2};\\\\q=\frac{b_2}{b_1}=\frac{\frac{1}{2}}{\frac{2}{3}}=\frac{3}{4};|q|<1\\\\S=\frac{b_1}{1-q}=\frac{\frac{2}{3}}{1-\frac{3}{4}}=\frac{8}{3}$

6) $a_1=6;a_2=14;14-6=22-14=8;\\\\d=a_2-a_1=14-6=8;\\\\S_n=\frac{2a_1+(n-1)*d}{2}*n;\\\\S_n=\frac{2*6+8(n-1)}{2}*n=(6+4n-4)n=(4n-2)n=4n^2-2n$