Mysterious Numerical Series

Here are a few series puzzles in which a number has mysteriously vanished, and your task is to restore it to its rightful place. Do not be deceived by the numerals. A sequence that contains numbers need not be mathematical in any serious sense. Arithmetic may make an appearance, but it may just as easily be a red herring. In puzzles of this sort, anything is fair game. No higher mathematics is required — only a flexible mind and a taste for looking at the obvious in an unobvious way. Try to find the simplest solution!

a) \(15\)    \(52\)    \(99\)    \(144\)    \(175\)    \(180\)    \(147\)    \(?\)    

b) \(11\)    \(12\)    \(26\)    \(79\)    \(81\)    \(163\)    \(491\)    \(492\)    \(?\)

c) \(3\)    \(23\)    \(229\)    \(2869\)    \(43531\)    \(?\)                           

d) \(0\)    \(5\)    \(8\)    \(8\)    \(2\)    \(3\)    \(5\)    \(2\)    \(9\)    \(4\)    \(?\)           

e) \(7\)    \(8\)    \(5\)    \(5\)    \(3\)    \(4\)    \(4\)    \(6\)    \(9\)    \(7\)    \(8\)    \(?\)

f) \(1\)    \(3\)    \(8\)    \(22\)    \(65\)    \(209\)    \(732\)    \(2780\)    \(?\)      

Answers 

a) \(64\). Take the difference of numbers (i.e., the first, second, and third differences), and you will see a pattern emerge. As shown below,

b) \(986\). The sequence follows this pattern: First, multiply by 1 and then add 1; next, multiply by 2 and then add 2; then, multiply by 3 and add 1; after that, multiply by 1 and add 2; next, multiply by 2 and add 1; followed by multiplying by 3 and adding 2. This sequence then repeats.

c) \(776887\). This sequence obeys the following pattern: \(2^{2}-1^{1}=3\), \(3^{3}-2^{2}=23\), \(4^{4}-3^{3}=229\), \(5^{5}-4^{4}=2869\), \(6^{6}-5^{5}=43531\), \(7^{7}-6^{6}=776887\), \(...\)

d) \(1\). The numbers shown are the digits following the decimal point when you divide \(1/17\)

e) \(8\). Each number is the number of letters in each month of the year, beginning with January.

f)\(11377\). This sequence obeys the following pattern: 

\(1^{1}=1\), 

\(2^{1}+1^{2}=3\), 

\(3^{1}+2^{2}+1^{3}=8\), 

\(4^{1}+3^{2}+2^{3}+1^{4}=22\), 

\(5^{1}+4^{2}+3^{3}+2^{4}+1^{5}=65\),  

\(6^{1}+5^{2}+4^{3}+3^{4}+2^{5}+1^{6}=209\), 

\(7^{1}+6^{2}+5^{3}+4^{4}+3^{5}+2^{6}+1^{7}=732\),                                                                     

\(8^{1}+7^{2}+6^{3}+5^{4}+4^{5}+3^{6}+2^{7}+1^{8}=2780\),    

\(9^{1}+8^{2}+7^{3}+6^{4}+5^{5}+4^{6}+3^{7}+2^{8}+1^{9}=11377\), 

\(...\)

References 

From Hoeflin, R. K. (1985, April). Problems:  43, 44, 45 & 48 in The Mega Test. Omni, 7(4), p. 132.