Fill in the blank

1, 4, 9, 16, 25, __________

The numbers are

\[1^2,2^2,3^2,4^2,5^2\]

\[\therefore\] Missing number

\[= 6^2 = 36\].

20, 19, 17, __________, 10, 5

The pattern is

- 1, - 2, ...

= 17 - 3 = 14.

6, 11, 21, 36. 56, __________

+ 5, + 10, + 15, + 20,...

= 56 + 25 = 81.

1, 6, 13, 22, 33, __________

+ 5. + 7, + 9, + 11,...

=33 + 13 = 46.

3, 9, 27, 81, __________

Each term of the given series is obtained by multiplying its preceding term by 3.

= 81 x 3 = 243.

5, 9, 17, 29. 45, __________

+ 4 + 8, + 12, + 16....

= 45 + 20 = 65.

3, 6, 18, 72, __________

x 2, x 3, x 4,...

= 72 x 5 = 360.

2, 15, 41, 80, __________

+ 13, + 26, + 39,....

= 80 + 52 = 132.

1, 2, 6, 24, __________

\[\therefore\] Missing number =

= 24 x 5 = 120.

2000, 1000, 500, 250, __________

Each term is divided by 2

\[= \frac{250}{2} = 125\].

34, 18, 10, 6, 4, __________

Each term is divided by 2 and then increased by 1 to obtain the next term.

\[\therefore\] Missing term

\[= \left(4\div 2\right)+1 = 3\]

462, 420, 380, __________, 306

- 42, - 40,...

= 380 - 38 = 342.

97, 86, 73, 58, 45, __________

- 11, - 13, - 15. - 13,...

\[therefore\] Missing number

= 45 - 11 = 34.

17, 19, 23, 29, __________, 37

The given series consists of consecutive prime numbers starting from 17.

The next prime number after 29 is 31.

So, the missing number is 31.

1, 4, 2, 8, 6, 24, 22, 88, __________

x 4, - 2, x 4, - 2,...

Missing number

= 88 - 2 = 86.

123456147, 12345614, 2345614, 234561, __________

The digits are removed one by one from the end as well as from the beginning in order so as to obtain the next term of the series.

Find out the wrong term.

1, 5, 10, 16, 20

Each term in the series is increased by 5 to obtain the next term.

So. 16 is wrong and must be replaced by (10 + 5) i.e. 15.

24, 27, 31, 33, 36

Each term in the series is increased by 3 to obtain the next term.

So. 31 is wrong and must be replaced by (27 + 3) i.e. 30.

196, 169, 144, 121, 80

The sequence is

\[\left(14\right)^2,\left(13\right)^2,\left(12\right)^2,\left(11\right)^2,\left(10\right)^2\]

So. 80 is wrong and must be replaced by

\[(10)^2\] i.e. 100.

3, 5, 7, 9, 11, 13

The series consists of consecutive prime numbers.

So. 9 is wrong.