2, 8, 18,?, 50, 72

T₅ = a + 4d, T₇ = a + 6d
? 5(a + 4d) = 7(a + 6d)
? 5a + 20d = 7a + 42d
? a = -11d
? T₁₂ = a + 11d = - 11d + 11d = 0
So, the twelfth term is 0.

The second and fourth term of the HP are 1/6 and 1/14 respectively,
Hence, for the corresponding AP, the second term is 6 and fourth term is 14.
Hence a + d = 6 and a + 3d = 14
? 2d = 8
? d = 4 and a = 2

Hence, the 10th term of this AP = a + (10 -1)d
= 2 + (10 - 1)x 4 = 2 + 9 x 4
= 2 + 36 = 38
Hence, for the corresponding HP, the 10th term is 1/38.

(a + b)/2 = 25
a + b = 50
?ab = 7
ab = 49
Hence, A can either be 7 or 49.
So, 49 is the answer.

Let term = l = ar^{n - 1} a = 5 and l = 20480
r = 20/5 = 4
? 20480 = 5 x (4)^n-1
(4)^n-1 = 20480/5 = 4096 = (4)⁶
n - 1 = 6
? n = 7

The nth term of a GP is ar^{n - 1} ,
5th term = ar^{5 - 1} = ar⁴ = 81
1st term = a = 16
? r⁴ = 81/16
? r = ?81/16 = 3/2
? 4th term = ar^{4 - 1} = ar³ = 16 x 3/2 x 3/2 x 3/2 = 54

Series pattern
1³ + 1, 3³ + 1, 4³ + 1, 5³ + 1
? Missing term = 6³ + 1 = 217

Series pattern
x 2 + 5, x 3 + 5, x 4 + 5, x 5 + 5,
? Missing term = 104 x 4 + 5 = 416 + 5 = 421

Series pattern
÷ 1, ÷ 2, ÷ 3, ÷ 4, ÷ 5
? Missing term = 840/1 = 840

Series pattern - 45, - 35, -25,...
? Missing term = 45 - 25 = 20

Series pattern +15, + 30, +60, +120
? Missing term = 49 + 60 = 109