The next term in the series is [tex]\frac{19}{6}[/tex]
Step-by-step explanation:
Given series ,
[tex]\frac{1}{2} ,\frac{1}{3} ,\frac{5}{6} ,\frac{7}{6} , 2[/tex]
- We have to find the next term in series.
To find the next term we first have to observe the pattern which is followed here.
=> The pattern followed is that a number is the sum of pervious 2 numbers.
It can be mathematically written as ,
[tex]n^{th} \ term = (n-1)^{th} \ term + (n-2)^{th} \ term[/tex]
=> Let us verify the pattern in the series,
[tex]3^{rd} \ term = 2^{nd} \ term + 1^{st} \ term[/tex]
[tex]\frac{5}{6} =\frac{1}{2} +\frac{1}{3} \\\\Taking \ LCM \ as \ 6\\\\\frac{5}{6} =\frac{3+2}{6} \\\\\frac{5}{6} =\frac{5}{6} \\\\[/tex]
Hence the pattern is verified.
=> Now finding the next term in the series,
[tex]6^{th} \ term = 5^{th} \ term + 4^{th} \ term[/tex]
[tex]6^{th} \ term =\frac{7}{6} +2 \\\\Taking \ LCM \ as \ 6\\\\\ 6^{th} \ term =\frac{7+12}{6} \\\\\ 6^{th} \ term =\frac{19}{6} \\\\[/tex]
Hence the next term is [tex]\frac{19}{6}[/tex]