Question

State the next term of each sequence a) 1, 16, 81, 256.... b) 1, 1, 2, 4, 7, 13, 24....

Answer 1

A)
$1,16,81,256,\underline{625}\\ \text{The general formula is } n^4 \text{ where } n\in \mathbb{N}$

b)
$1,1,2,4,7,13,24,\underline{44}\\ \text{Each number is the sum of three preceding numbers.}$

Answer 2

A) 1, 16, 81, and 256 are all perfect squares. Find their squares:
1, 4, 9, 14.
In this sequence, you first at 3, then 5, then 7. So, logically, you would add 9 next. 14 + 9 is 23, and 23 squared is 529.
b) I'm not sure the data points for this sequence are correct. I'd have to guess 42 based on this difference chart:
1    1    2     4     7     13     24   42
  +0  +1  +2    +3   +6   +11  +18
      +1  +1  +1    +3   +5   +9
         +0  +0   +2   +2   +4
            +0   +2  +0   +2