a.
[tex]\frac{(n+1)!}{n!} =42\\\\\frac{n!(n+1)}{n!} =42\\\\n+1=42\\\\n=41[/tex]
b.
[tex]\frac{(n+2)!}{n!} =6\\\\\frac{n!(n+1)(n+2)}{n!} =6\\\\(n+1)(n+2)=6\\\\n^2+3n+2=6\\\\n^2+3n-4=0\\\\\Delta=9+16=25\\\\n_1=\frac{-3+5}{2} =1\\\\n_2=\frac{-3-5}{2} =-4 < 0\ nu[/tex]
c.
[tex]\frac{(n+2)!-n!}{n!} =3n+5\\\\\frac{n!(n+1)(n+2)-n!}{n!} =3n+5\\\\(n+1)(n+2)-1=3n+5\\\\n^2+3n+2-1=3n+5\\\\n^2=4\\n=2[/tex]
d.
[tex]\frac{(n+3)!}{(n+1)!} =90\\\\\frac{(n+1)!(n+2)(n+3)}{(n+1)!} =90\\\\(n+2)(n+3)=90\\\\n^2+5n-84=0\\\\\Delta=25+336=361\\\\n=\frac{-5+19}{2} =7[/tex]
