by Bill Graham
Look for patterns in the sequences below. Try to list the next five terms in each of the sequences. My answers follow, though there may be other rules that result in the same sequence.
a) 1, 3, 6, 10, . . .
b) 2, 3, 5, 7, 11,13, . . .
c) O, T, T, F, F, S, S, E, . . .
d) 1, 4, 9, 16, . . .
e) 1, 1, 2, 3, 5, 8, 13, . . .
f) 1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 21, . . .
g) 2, 3, 5, 9, 17, . . .
h) 1, 2, 6, __ , 120, 720, . . .
i) 2, 10, 30, 68, 130, 222, . . .
j) 12, 1, 1, 1, 2, 1, 3, . . .
a) The first pair differs by two, the second pair by three, etc.
The next five terms are 15, 21, 28, 36 and 45.
b) This is a sequence of prime numbers.
The next five are 17, 19, 23, 29 and 31.
c) This sequence is the first letters of the numbers from one to eight.
After E for eight comes N, T, E, T and T.
d) These are the perfect square numbers.
The next terms are 25, 36, 49, etc.
e) This is called the Fibonocci sequence.
Each new term is obtained by adding the previous two.
The next term is 21 obtained by adding eight and 13.
After that come 34, 55, 89 and 144.
f) This pattern is one followed by a gap of one.
Then the next two consecutive digits followed by a gap of two digits, etc.
The next terms are 22, 23, 24, 25 and 31.
g) The difference between the first two terms is one, then two,
then four, then eight, etc. The next terms are 33, 65, 129, etc.
Another way to look at this is the sequence of 2n + 1.
Or, Barbara’s method is to multiply the previous term by two
and subtract one to get the next term.
h) The first term times two gives the second term.
The second term times three gives the third term, etc.
The next terms are 24, 120, 720, 5,040, 40,320, etc.
If you ever studied permutations and com binations,
you will recognize this sequence as 1!, 2!, 3!, 4!, etc.
i) This was hard for me to figure out and harder to explain.
The difference between each pair of terms makes the new sequence
8, 20, 38, 62, etc.
The difference between each of those terms makes the sequence
12, 18, 24, 30, . . ., which has a common difference of six.
Working backward from there, the next terms are 350, 520, 738, 1,010, . . .
Since I saw this sequence somewhere and figured it out myself,
my confidence was shaken a bit when a friend got a different answer.
After much effort, I found a function which will produce the terms
in the answer above, namely: x3 + 3x2 + 4x + 2.
j) This pattern is the number of chimes a clock would make
if it chimes once on the half hour starting at 12 o'clock.
The next five terms are 1,4, 1, 5 and 1.
Vol 3, No 1 Answers
