(a) The pattern 1, 4, 9, etc consists of numbers that are perfect
squares. In other words each number in the list is the product of
a number with itself. In fact, the number that is squared is just the
position of the entry in the list: so the first number 1 is 12
= 1 x 1, the second number 4 is 22 = 2 x 2, the third number 9
is 32 = 3 x 3, and so on. The three blanks should be filled
in with 62, 72, and 82, or 36, 49 and
64.
(b) This sequence of numbers starts with 3, 4, and after that, each
number is the sum of the previous two numbers (you might be familiar
with the Fibonacci sequence which is like this one except that it starts
with 1, 1 instead of 3, 4). Anyhow, the next three numbers in the
sequence are 18 + 29 = 47, followed by 29 + 47 = 76, followed by 47 + 76
= 123.
(c) This is a tricky sequence -- starting with the number 5, to get the
next number in the sequence you alternately double (multiply by 2)
followed by subtracting 1. Since the last given number 33 comes from
subtracting 1 from the previous number 34, to get the next term we should
multiply by 2. So the next term is 66, then we subtract 1 to get 65 to
fill in the other blank.
(d) In this sequence, to go from one number to the next, you multiply by
the "next number" (or by the position of the new number in the
sequence). So to go from 1 to 2, you multiply by 2, from 2 to 6 you
multiply by 3, and so forth. Another way to say this is that each element
in the sequence is the product of all the numbers from 1 up to the
position of the number in the sequence. So the fourth number, 24, is the
product 1 x 2 x 3 x 4 of the numbers from 1 to 4. And 120 is 24 x 5. So
the next term should be 120 x 6 = 720, the one after that should be 720 x
7 = 5040, and the last blank should contain 5040 x 8 = 40320. These
numbers are often called "factorial numbers", or "factorials". For
instance 120 = 5 x 4 x 3 x 2 x 1 is called "5 factorial", and it is
written 5!
