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# Find the missing variable from $$a, d, n$$ and $$a_n$$, where a is the first term, d is the common difference and $$a_n$$ is the nth term of AP. (i) $$a = 7, d = 3, n = 8$$ (ii) $$a = –18, n = 10, a_n = 0$$ (iii) $$d = –3, n = 18, a_n = -5$$ (iv) $$a = –18.9, d = 2.5, a_n = 3.6$$ (v) $$a = 3.5, d = 0, n = 105$$

(i) a = 7, d = 3, n = 8

We need to find $$a_n$$ here.

Using formula $$a_n = a + (n - 1)d$$

Putting values of a, d and n,

$$a_n$$ = 7 + (8 - 1) 3
= 7 + (7) 3 = 7 + 21 = 28

(ii) a = –18, n= 10, $$a_n = 0$$

We need to find d here.

Using formula $$a_n = a + (n - 1)d$$

Putting values of a,$$a_n , n$$

$$\Rightarrow$$ 0 = –18 + (10 – 1) d
$$\Rightarrow$$ 0 = -18 + 9d
$$\Rightarrow$$ 18 = 9d
$$\Rightarrow$$ d = 2

(iii) d = –3, n = 18, $$a_n = -5$$

We need to find a here.

Using formula $$a_n = a + (n - 1)d$$

Putting values of d, $$a_n , n$$

$$\Rightarrow$$ –5 = a + (18 – 1) (–3)
$$\Rightarrow$$ - 5 = a + (17) (-3)
$$\Rightarrow$$ -5 = a – 51
$$\Rightarrow$$ a = 46

(iv) a = –18.9, d = 2.5, $$a_n = 3.6$$

We need to find n here.

Using formula $$a_n = a + (n - 1)d$$

Putting values of $$d, a_n , n$$

$$\Rightarrow$$ 3.6 = –18.9 + (n – 1) (2.5)
$$\Rightarrow$$ 3.6 + 18.9 = (n - 1)(2.5)
$$\Rightarrow$$ 22.5 = (n -1)(2.5)
$$\Rightarrow$$ (n - 1) = $$\frac{22.5}{2.5}$$
$$\Rightarrow$$ n - 1 = 9
$$\Rightarrow$$ n = 10

(v) a = 3.5, d = 0, n = 105

We need to find $$a_n$$ here.

Using formula $$a_n = a + (n - 1)d$$

Putting values of d, n and a,

$$\Rightarrow a_n$$ = 3.5 + (105 - 1) (0)
$$\Rightarrow$$$$a_n = 3.5 + 104 × 0$$
$$\Rightarrow$$ $$a_n = 3.5 + 0$$
$$\Rightarrow a_n = 3.5$$