2. Which of the following are terms of the series with nth term T-3n +17? a) 80 b) 170 c)217 d) 312 e) 278 f) 3566

Answer:The correct options are a,b,e and f.Step-by-step explanation:It is given that the nth terms of the series is defined as[tex]T_n=3n+17[/tex]Subtract 17 from both the sides.[tex]T_n-17=3n[/tex]Divide both sides by 3.[tex]\frac{T_n-17}{3}=n[/tex]The term Tₙ is a term of given series if n is a positive integer.(a) The given term is 80.[tex]n=\frac{80-17}{3}=21[/tex]Since n is a positive integer, therefore 80 is a term of given series.(b) The given term is 170.[tex]n=\frac{170-17}{3}=51[/tex]Since n is a positive integer, therefore 170 is a term of given series.(c) The given term is 217.[tex]n=\frac{217-17}{3}=66.67[/tex]Since n is not a positive integer, therefore 217 is a term of given series.(d) The given term is 312.[tex]n=\frac{312-17}{3}=98.33[/tex]Since n is not a positive integer, therefore 312 is a term of given series.(e) The given term is 278.[tex]n=\frac{278-17}{3}=87[/tex]Since n is a positive integer, therefore 278 is a term of given series.(f) The given term is 3566.[tex]n=\frac{3566-17}{3}=1183[/tex]Since n is a positive integer, therefore 3566 is a term of given series.Thus, the correct options are a, b, e and f.