1, 2, 3, 6, 9, 18,?, 54

The series consists of squares and cubes of consecutive natural numbers i.e. 1², 1³, 2², 2³, 3², 3³, 4², .....

So, missing term = 4³ = 64.

The pattern is - 45, - 35, - 25, .....

So, missing term = 20 - 15 = 5.

Clearly, 2 x 3 = 6, 6 x 3 = 18, 18 x 3 = 54,.....

So, the series is a G.P. in which a = 2, r = 3.

Therefore 8th term = ar^8-1 = ar⁷ = 2 x 3⁷ = (2 x 2187) = 4374.

Each term in the series is obtained by adding 1 to the square of the preceding term.

So, missing term = (101)² + 1 = 10202.

The digits are removed one by one from the beginning and the end in order alternately

so as to obtain the subsequent terms of the series.

The pattern is x 2 + 1, x 2 - 1, x 2 + 1, x 2 - 1,.....

So, missing term = 101 x 2 + 1 = 203.

The pattern is + 1, + 3, + 6,....., i.e. + 1, + (1, + 2), + (1 + 2 + 3),.....

So, missing term = 15 + (1 + 2 + 3 + 4) = 25.

The sum of any three consecutive terms of the series gives the next term.

So, missing number = 8 + 15 + 27 = 50.

The given sequence is a combination of two series :

I. 3, 7, 13, 21, 31, ? and II. 4, 7, 13, 22, 34

The pattern in I is + 4, + 6, + 8, + 10,.....

The pattern in II is + 3, + 6, + 9, + 12,.....

So, missing term = 31 + 12 = 43.

The pattern is - 4, - 9, - 16,.....i.e. - 2², - 3², - 4²,.....

So, missing pattern = 169 - 5² = 169 - 25 = 144.