406 McGRAW-HILL’S SAT
Analyzing Sequences
Asequenceis just a list of numbers, each of which is
called a term. An SAT math question might ask you to
use a sequential pattern to solve a problem, such as
“How many odd numbers are in the first 100 terms of
the sequence 1, 2, 3, 1, 2, 3,... ?”
An SAT sequence question usually gives you
the first few terms of a sequence or the rule for
generating the sequence, and then asks you ei-
ther to find a specific termin the sequence (as
in “What is the 59th term of this sequence?”)
or to analyze a subset of the sequence(as in
“What is the sum of the first 36 terms of this se-
quence?”). To tackle sequence problems:- Use the pattern or rule to write out the first
six to eight terms of the sequence. - Try to identify the pattern in the sequence.
Notice in particular when the sequence be-
gins to repeat itself, if it does. - Use this pattern, together with whole-num-
ber division (Chapter 7, Lesson 7), if it’s
helpful, to solve the problem.
Example:
1, 2, 2, ...The first three terms of a sequence are shown above.
Each term after the second term is found by dividing
the preceding term by the term before that. For exam-
ple, the third term is found by dividing the second
term, 2, by the first term, 1. What is the value of the
218th term of this sequence?
Don’t panic. You won’t have to write out 218 terms!
Just write out the first eight or so until you notice that
the sequence begins to repeat. The fourth term is
2 ÷ 2 = 1, the fifth term is 1 ÷2 = 1/2, and so
on. This gives the sequence 1, 2, 2,1,^1 / 2 ,^1 / 2 ,
1, 2,....
Notice that the first two terms of the sequence,
1 and 2, have come back again! This means that
the first six terms in the sequence, the underlined
ones, will just repeat over and over again. Therefore,
in the first 218 terms, this six-term pattern will repeat
times, or 36 with a remainder of 2. So,
the 218th term will be the same as the second term in
the sequence, which is 2.
Example:
1, 1, 0, 1, 1, 0,...
If the sequence above repeats as shown, what is the
sum of the first 43 terms of this sequence?Since the sequence clearly repeats every three
terms, then in 43 terms this pattern will repeat 43 ÷ 3
= 14 (with remainder 1) times. Each full repetition of
the pattern 1, 1, 0 has a sum of 0, so the first 14 rep-
etitions have a sum of 0. This accounts for the sum of
the first 14 3 = 42 terms. But you can’t forget the
“remainder” term! Since that 43rd term is 1, the
sum of the first 43 terms is 1.You won’t need to use the formulas for “arith-
metic sequences” or “geometric sequences”
that you may have learned in algebra class.
Instead, SAT “sequence” questions simply
require that you figure out the pattern in the
sequence.218 6 36÷=^23
Lesson 1: Sequences