125,80,45,20,?

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 - 21, - 19, - 17, - 15,.....

So, missing term = 48 - 13 = 35.

P @ Q ? P is the wife of Q ...(1)

Q $ T ? Q is the brother of T ...(2)

T # U ? T is the daughter of U

Hence, ? Q is the son of U ...(3)

U * W ? U is the father of W.

From (1) and (3), U is the father-in-law of P.

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 x 2, x 3/2, x 2, x 3/2, x 2,.....

So, missing term = 18 x 3/2 = 27.

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.