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
![T_n=3n+17](https://tex.z-dn.net/?f=T_n%3D3n%2B17)
Subtract 17 from both the sides.
![T_n-17=3n](https://tex.z-dn.net/?f=T_n-17%3D3n)
Divide both sides by 3.
![\frac{T_n-17}{3}=n](https://tex.z-dn.net/?f=%5Cfrac%7BT_n-17%7D%7B3%7D%3Dn)
The term Tₙ is a term of given series if n is a positive integer.
(a) The given term is 80.
![n=\frac{80-17}{3}=21](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B80-17%7D%7B3%7D%3D21)
Since n is a positive integer, therefore 80 is a term of given series.
(b) The given term is 170.
![n=\frac{170-17}{3}=51](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B170-17%7D%7B3%7D%3D51)
Since n is a positive integer, therefore 170 is a term of given series.
(c) The given term is 217.
![n=\frac{217-17}{3}=66.67](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B217-17%7D%7B3%7D%3D66.67)
Since n is not a positive integer, therefore 217 is a term of given series.
(d) The given term is 312.
![n=\frac{312-17}{3}=98.33](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B312-17%7D%7B3%7D%3D98.33)
Since n is not a positive integer, therefore 312 is a term of given series.
(e) The given term is 278.
![n=\frac{278-17}{3}=87](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B278-17%7D%7B3%7D%3D87)
Since n is a positive integer, therefore 278 is a term of given series.
(f) The given term is 3566.
![n=\frac{3566-17}{3}=1183](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B3566-17%7D%7B3%7D%3D1183)
Since n is a positive integer, therefore 3566 is a term of given series.
Thus, the correct options are a, b, e and f.