Find the sums given below :

(i) $7 + 10\dfrac{1}{2} + 14 + ……. + 84$

(ii) 34 + 32 + 30 + ……. + 10

(iii) -5 + (-8) + (-11) + …… + (-230)

(i) Given,

⇒ $7 + 10\dfrac{1}{2} + 14 + ……. + 84$

⇒ 7 + 10.5 + 14 + ……… + 84

The above sequence is an A.P. with first term (a) = 7 and common difference (d) = 10.5 - 7 = 3.5

Let nth term be 84.

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ 84 = 7 + (n - 1)(3.5)

⇒ 84 = 7 + 3.5n - 3.5

⇒ 84 = 3.5n + 3.5

⇒ 3.5n = 84 - 3.5

⇒ 3.5n = 80.5

⇒ n = $\dfrac{80.5}{3.5}$ = 23.

By formula,

S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$\Rightarrow S_{23} = \dfrac{23}{2}[2 \times 7 + (23 - 1) \times 3.5] \\[1em] = \dfrac{23}{2}[14 + 22 \times 3.5] \\[1em] = \dfrac{23}{2}[14 + 77] \\[1em] = \dfrac{23}{2} \times 91 \\[1em] = \dfrac{2093}{2} \\[1em] = 1046\dfrac{1}{2}.$

Hence, sum = $1046\dfrac{1}{2}$.

(ii) Given,

34 + 32 + 30 + ……. + 10

The above sequence is an A.P. with first term (a) = 34 and common difference (d) = 32 - 34 = -2.

Let nth term be 10.

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ 10 = 34 + (n - 1)(-2)

⇒ 10 = 34 - 2n + 2

⇒ 10 = 36 - 2n

⇒ 2n = 36 - 10

⇒ 2n = 26

⇒ n = $\dfrac{26}{2}$ = 13.

By formula,

S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$\Rightarrow S_{13} = \dfrac{13}{2}[2 \times 34 + (13 - 1) \times -2] \\[1em] = \dfrac{13}{2}[68 + 12 \times -2] \\[1em] = \dfrac{13}{2}[68 - 24] \\[1em] = \dfrac{13}{2} \times 44 \\[1em] = 13 \times 22 \\[1em] = 286.$

Hence, sum = 286.

(iii) Given,

-5 + (-8) + (-11) + ……. + (-230)

The above sequence is an A.P. with first term (a) = -5 and common difference (d) = -8 - (-5) = -8 + 5 = -3.

Let nth term be -230.

By formula,

a_n = a + (n - 1)d

Substituting values we get :

⇒ -230 = -5 + (n - 1)(-3)

⇒ -230 = -5 - 3n + 3

⇒ -230 = -3n - 2

⇒ 3n = -2 + 230

⇒ 3n = 228

⇒ n = $\dfrac{228}{3}$ = 76.

By formula,

S_n = $\dfrac{n}{2}[2a + (n - 1)d]$

Substituting values we get :

$\Rightarrow S_{76} = \dfrac{76}{2}[2 \times -5 + (76 - 1) \times -3] \\[1em] = 38 \times [-10 + 75 \times -3] \\[1em] = 38 \times [-10 - 225] \\[1em] = 38 \times -235 \\[1em] = -8930.$

Hence, sum = -8930.