Random Puzzles 

Workings

Hint
The sequence involves both adding and multiplying in some way.

Answer
153.

The sequence is created using: +1 x1 +2 x2 +3 x3 +4 x4 +5.

Workings

Hint
The word FEBRUARY appears 21 times.

Answer
The word FEBRUARY appears 21 times.

Workings

Hint
A little algebra might help.

Answer
¹/₈.

Reasoning
If we convert the question to algebra, we have:
((n^2 + 4) x 2 − 3) x 4 − n = 20

Expanding the brackets and simplifying gives:

(2n^2 + 8 − 3) x 4 − n = 20

    (2n^2 + 5) x 4 − n = 20

         8n^2 + 20 − n = 20

              8n^2 − n = 0 (*)

                8n − 1 = 0

                    8n = 1

                     n = ¹/₈

In the equation marked (*) zero is also a potential solution, but as the question tells us that we "Start with a number larger than 0" we know that n can't be 0, and therefore we can safely divide by n.

Double-Checking
                 ¹/₈
   squared     = ¹/₆₄
   add 4       = 4¹/₆₄
   double      = 8¹/₃₂
   take away 3 = 5¹/₃₂
   times 4     = 20¹/₈

   subtracting ¹/₈ gives 20, as required.

Workings

Answer
6: the numbers denote the number of letters each month has starting with January, therefore August has 6 letters.