10, 100, 200, 310,?

The given sequence is a combination of two series :

I. 625, 125, 25, 5 and II. 5, 25, ?

The pattern in I is ÷ 5, while that in II is x 5. So, missing term = 25 x 5 = 125.

The pattern is + 36, + 60, + 90,.....i.e. + [6 x (6 + 0)], + [6 x (6 + 4)], + [6 x (6 + 9)],...

So, missing term = 210 + [6 x (6 + 15)] = 210 + 126 = 336.

Let the missing terms of the series be x₁ and x₂.

Thus, the sequence 20, 20, 19, 16, 17, 13, 14, 11, xv x2 is a combination of two series :

I. 20, 19, 17, 14, x₁ and II. 20, 16, 13, 11, x₂

The pattern in I is - 1, - 2, - 3,......So, missing term, x₁ = 14 - 4 = 10.

The pattern in II is - 4, - 3, - 2,......So, missing term, x₂ = 11 - 1 = 10.

The pattern is + 11, + 22, + 33, .....

So, missing term = 72 + 44 = 116.

The pattern is x 1 + 1, x 2 + 2, x 3 + 3, x 4 + 4,.....

So, missing term = 112 x 5 + 5 = 565.

The pattern is - 1, x 10 + 1, - 1, x 10 + 1, - 1, x 10 + 1, .....

So, missing term = 10 x 10 + 1 = 101.

The pattern is + 5, + 20, + 80, + 320, ...... i.e. + (5 x 1²), + (5 x 2²), + (5 x 4²), + (5 x 8²),.....

So, missing term = 427 + (5 x 16²) = 427 + 1280 = 1707.

In these 2 positions one common face with number 1 is in the same position. Hence according to the rule number (3), 2 is opposite 6 and 3 is opposite to 5. Therefore opposite to 4 is 1.

The symbols of the adjacent faces to the face with symbol * are @, -, + and $. Hence the required symbol is 8.

Here the common faces with number 3, are in same positions. Hence 6 is opposite to 2 and 5 is opposite to 1. Therefore 4 is opposite to 3.