Correct Answer - Option 3 : 1649

**Concept:**

Let us consider sequence a_{1}, a_{2}, a_{3} …. a_{n} is an A.P.

- Common difference “d”= a
_{2} – a_{1} = a_{3} – a_{2} = …. = a_{n} – a_{n – 1}

- n
^{th} term of the A.P. is given by a_{n} = a + (n – 1) d
- n
^{th} term from the last is given by a_{n} = l – (n – 1) d
- sum of the first n terms = S = n/2[2a + (n − 1) × d] Or sum of the first n terms = n/2(a + l)

Where, a = First term, d = Common difference, n = number of terms and a_{n} = n^{th} term

__Calculation:__

Given: nth term of an A.P = a_{n} = \(\frac{{2\; +\; {\rm{n}}}}{3}\)

For first term, put n = 1

a_{1} = a = (2 + 1)/3 = 3/3 = 1

For last term, put n = 97

l = (97 + 2)/3 = 99/3 = 33

We have to find the sum of first 97 terms,

S = (97/2) (1 + 33) (∵ S = n/2(a + l))

S = 97 × 17 = 1649