Correct Answer - Option 3 : 1649
Concept:
Let us consider sequence a1, a2, a3 …. an is an A.P.
- Common difference “d”= a2 – a1 = a3 – a2 = …. = an – an – 1
- nth term of the A.P. is given by an = a + (n – 1) d
- nth term from the last is given by an = 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 an = nth term
Calculation:
Given: nth term of an A.P = an = \(\frac{{2\; +\; {\rm{n}}}}{3}\)
For first term, put n = 1
a1 = 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