(i) For this given A.P., 7 + \(10 \frac{1}{2}\) + 14 + ...... + 84

First term, a = 7

n^th term, a_n = 84

Common difference, d = a₂ - a₁

= \(10 \frac{1}{2} - 7\) = \( \frac{21}{2} - 7\) = \(\frac{7}{2}\)

Let 84 be the n^th term of this A.P., then as per the n^th term formula,

a_n = a(n-1)d

84 = 7+(n – 1)×7/2

77 = (n-1)×7/2

22 = n−1

n = 23

We know that, sum of n term is;

S_n = n/2 (a + l) , l = 84

S_n = 23/2 (7+84)

S_n = (23×91/2) = 2093/2

S_n = \(1046 \frac{1}{2}\)

(ii) Given, 34 + 32 + 30 + ……….. + 10

first term, a = 34

common difference, d = a₂−a₁ 

= 32 - 34 = -2

n^th term, a_n= 10

Let 10 be the n^th term of this A.P.,

a_n= a +(n−1)d

10 = 34+(n-1)(-2)

-24 = (n -1)(-2)

12 = n -1

n = 13

We know that, sum of n terms is;

S_n = n/2 (a +l) , l = 10

= 13/2 (34 + 10)

= (13×44/2) = 13 × 22

= 286

(iii) Given, (−5) + (−8) + (−11) + ………… + (−230)

For this A.P.,

First term, a = −5

nth term, a_n= −230

Common difference, d = a₂−a₁ 

= (−8)−(−5)

d = − 8+5 = −3

Let −230 be the n^th term of this A.P., and by the n^th term formula we know,

a_n= a+(n−1)d

−230 = − 5+(n−1)(−3)

−225 = (n−1)(−3)

(n−1) = 75

n = 76

And, Sum of n term,

S_n = n/2 (a + l)

= 76/2 [(-5) + (-230)]

= 38(-235)

= -8930