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# Find the number of terms in each of the following APs: (i) 7, 13, 19…., 205 (ii) 18,$${15}{\dfrac{1}{2}}$$ , 13…, -47

(i) 7, 13, 19 …, 205

First term = a = 7,
Common difference = d
= 13 – 7 = 19 – 13 = 6

And $$a_n = 205$$

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

$$\Rightarrow$$ 205 = 7 + (n - 1) 6 = 7 + 6n – 6
$$\Rightarrow$$ 205 = 6n + 1
$$\Rightarrow$$ 204 = 6n
$$\Rightarrow$$ n = 34

Therefore, there are 34 terms in the given arithmetic progression.

(ii) 18, $${15}{\dfrac{1}{2}}$$ , 13 …, -47

First term = a =18,
Common difference = d
= $${15}{\dfrac{1}{2}} - 18$$
$$= {{31} \over {2}} - 18$$
$$= {{31 - 36} \over {2}}$$
$$= {{-5} \over {2}}$$

And $$a_n = -47$$

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

$$\Rightarrow$$ 47 = 18 + (n - 1) $${{-5} \over {2}}$$
$$\Rightarrow$$ 36 - $${{5} \over {2}}$$ n + $${{5} \over {2}}$$
$$\Rightarrow$$ -94 = 36 - 5n + 5
$$\Rightarrow$$ 5n = 135
$$\Rightarrow$$ n = 27

Therefore, there are 27 terms in the given arithmetic progression.