Answer:
sequence 1 : arithmetic , sequence 2 : geometric
Step-by-step explanation:
• An arithmetic sequence has a common difference d , between consecutive terms.
• A geometric sequence has a common ratio r , between consecutive terms
sequence 1
[tex]\frac{1}{2}[/tex] , [tex]\frac{7}{6}[/tex] , [tex]\frac{11}{6}[/tex] , [tex]\frac{5}{2}[/tex]
Check for common difference , d
[tex]\frac{7}{6}[/tex] - [tex]\frac{1}{2}[/tex] = [tex]\frac{7}{6}[/tex] - [tex]\frac{3}{6}[/tex] = [tex]\frac{4}{6}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{11}{6}[/tex] - [tex]\frac{7}{6}[/tex] = [tex]\frac{4}{6}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{5}{2}[/tex] - [tex]\frac{11}{6}[/tex] = [tex]\frac{15}{6}[/tex] - [tex]\frac{11}{6}[/tex] = [tex]\frac{4}{6}[/tex] = [tex]\frac{2}{3}[/tex]
Since there is a common difference of d = [tex]\frac{2}{3}[/tex] , then sequence is arithmetic
sequence 2
[tex]\frac{1}{2}[/tex] , [tex]\frac{1}{3}[/tex] , [tex]\frac{2}{9}[/tex] , [tex]\frac{4}{27}[/tex]
check for common difference , d
[tex]\frac{1}{3}[/tex] - [tex]\frac{1}{2}[/tex] = [tex]\frac{2}{6}[/tex] - [tex]\frac{3}{6}[/tex] = - [tex]\frac{1}{6}[/tex]
[tex]\frac{2}{9}[/tex] - [tex]\frac{1}{3}[/tex] = [tex]\frac{2}{9}[/tex] - [tex]\frac{3}{9}[/tex] = - [tex]\frac{1}{9}[/tex]
It is obvious there is no common difference
check for common ratio , r
[tex]\frac{\frac{1}{3} }{\frac{1}{2} }[/tex] = [tex]\frac{1}{3}[/tex] × [tex]\frac{2}{1}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{\frac{2}{9} }{\frac{1}{3} }[/tex] = [tex]\frac{2}{9}[/tex] × [tex]\frac{3}{1}[/tex] = [tex]\frac{6}{9}[/tex] = [tex]\frac{2}{3}[/tex]
[tex]\frac{\frac{4}{27} }{\frac{2}{9} }[/tex] = [tex]\frac{4}{27}[/tex] × [tex]\frac{9}{2}[/tex] = [tex]\frac{36}{54}[/tex] = [tex]\frac{2}{3}[/tex]
There is a common ratio of r = [tex]\frac{2}{3}[/tex] , then sequence is geometric
