# The nth term of an A.P is $\frac{{2\; + \;{\rm{n}}}}{3}$, then the sum of first 97 terms is

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The nth term of an A.P is $\frac{{2\; + \;{\rm{n}}}}{3}$, then the sum of first 97 terms is

1. 1648
2. 1561
3. 1649
4. 1751
5. None of these

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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