(i) Given A.P. : 9, 17, 25, …

First term (a) = 9, common difference (d) 

= 17 – 9 = 8

Let no. of terms be n

S_n = 636

⇒ n/2[2a + (n – 1)d] = 636

⇒ n/2[2 × 9+ (n – 1)8] = 636

⇒ n/2[18 + 8n – 8] = 636

⇒ n/2[8n + 10] = 636

⇒ n(4n + 5) = 636

⇒ 4n² + 5n = 636

⇒ 4n² + 5n – 636 = 0

⇒ 4n² + 53 n – 48n – 636 = 0

⇒ n(4n + 53) – 12(4n + 53) = 0

⇒ (4n + 53)(n – 12) = 0

⇒ n – 12 = 0 or 4n + 53 = 0

⇒ n = 12 or – 53/4

∵ n cannot be  negative

So, ignore n = – 53/4

∴ n = 12

Hence, sum of 12 terms of given A.P. is 636.

(ii) Given A.P. 63, 60, 57 …..

First term(a) = 63

Common difference (d) = 60 – 63 = -3

Let number of terms be n

S_n = 693

We know that

S_n = n/2[2a + (n – 1)d]

⇒ 693 = n/2[2 × 63 + (n – 1) (-3)]

⇒ 693 = n/2[126 – 3n + 3]

⇒ 1386 = n(129 – 3n)

⇒ 1386 = 129n – 3n²

⇒ 3n² – 129n + 1386 = 0

⇒ n² – 43n + 462 = 0

⇒ n² – 21n – 22n + 462 = 0

⇒ = n(n – 21) – 22(n – 21) = 0

⇒ (n – 21)(n – 22) = 0

⇒ n – 21 = 0 or n – 22 = 0

n = 21 or n = 22

By taking 21 or 22 terms of given A.P. we will get sum 693.