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# Find the sums given below: (i) $$7 + {10}{\dfrac{1}{2}} + 14 + .......+ 84$$ (ii) 34 + 32 + 30 + .....… + 10 (iii) –5 + (–8) + (–11) + …..... + (–230)

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

Here First term = a = 7,
Common difference = d = $${{21} \over {2}} - 7 = {{21 - 15} \over {2}} = {{7} \over {2}} = 3.5$$

And Last term = l = 84

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula $$a_n = a + (n - 1)d$$ , to find nth term of arithmetic progression,

$$\Rightarrow$$ [7 + (n - 1) (3.5)] = 84
$$\Rightarrow$$ 7 + (3.5) n - 3.5 = 84
$$\Rightarrow$$ 3.5n = 84 + 3.5 – 7
$$\Rightarrow$$ 3.5n = 80.5
$$\Rightarrow$$ n = 23

Therefore, there are 23 terms in the given AP.

It means n = 23.

Applying formula $$S_n = {{n} \over {2}} (a + l)$$, to find sum of n terms of AP,

$$\Rightarrow S_n = {{23} \over {2}} (7 + 84)$$
$$\Rightarrow S_{23} = {{23} \over {2}} (91) = 1046.5$$

(ii) 34 + 32 + 30 + … + 10
Here First term = a = 34,
Common difference = d = 32 – 34 = –2
And Last term = l = 10

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula $$a_n = a + (n - 1)d$$, to find nth term of arithmetic progression,

$$\Rightarrow$$ [34 + (n - 1) (-2)] = 10
$$\Rightarrow$$ 34 – 2n + 2 = 10
$$\Rightarrow$$ 2n = 26
$$\Rightarrow$$ n = 13

Therefore, there are 13 terms in the given AP.

It means n = 13.

Applying formula $$S_n = {{n} \over {2}} (a + l)$$, to find sum of n terms of AP,

$$S_{13} = {{13} \over {2}} (34 + 10)$$
$$= {{13} \over {2}} (44) = 286$$

(iii) -5 + (-8) + (-11) + … + (-230)

Here First term = a = –5,
Common difference = d
= –8 – (–5) = –8 + 5 = –3

And Last term = l = -230

We do not know how many terms are there in the given AP.

So, we need to find n first.

Using formula $$a_n = a + (n - 1)d$$, to find nth term of arithmetic progression,

$$\Rightarrow$$ [-5 + (n - 1) (-3)] = -230
$$\Rightarrow$$ -5 - 3n + 3 = -230
$$\Rightarrow$$ -3n = -228
$$\Rightarrow$$ n = 76

Therefore, there are 76 terms in the given AP.

It means n = 76.

Applying formula $$S_n = {{n} \over {2}} (a + l)$$, to find sum of n terms of AP,
$$S_{76} = {{76} \over {2}} (-5 - 230)$$
$$= 38 (-235) = -8930$$