J. Progressions
Sequences
and Progressions
An ordered list
of numbers is called a sequence. For example, a sequence of positive
even numbers would be:
2, 4, 6, 8, 10,
…
Within a sequence,
each individual member is called a term. The terms are defined
by their position in the sequence. For example, in the above
sequence, 2 is the first term, 4 is the second term, and 6 is
the third term. The ellipsis symbol (…) indicates that the
sequence continues beyond the terms that are written. For the
above sequence, the next term would be 12, then 14, and so on.
Even though these two terms (and those beyond) are not written
out, we know that they are terms within the sequence because
the ellipsis indicates that the sequence continues forever (to
infinity).
A common kind
of sequence problem that you may encounter on the GMAT is one
in which you are given a few terms of a sequence and asked to
define the next term. In the above example, given that the sequence
was defined as consisting of positive even numbers, it was easy
to deduce that the next two terms in the sequence are 12 and
14. Often however, sequences will be more complicated. Given
more complex sequences, how can you determine the next term in
a sequence of numbers?
The key to solving
these problems is to determine the relationship between the terms
in the sequence that you are given. This relationship can be
described in terms of a progression, a function or manipulation
that can be applied to each individual term of a sequence that
will generate the next term in that sequence.
The types of
progressions typically found on the GMAT can often be described
as being either arithmetic or multiplicative. Let's now look
at examples of each of these.
Arithmetic progressions
Simply stated,
in an arithmetic progression, a fixed amount is added to each
term in order to generate the next term. An important consequence
of this is that the difference between any two consecutive terms
will remain constant for the entire sequence.
Here is an example
of an arithmetic sequence:
0, 3, 6, 9, 12,
15,…
What is the constant
value, or fixed amount, being added to each term to generate
the next? We can figure this out by subtracting any term in the
sequence from the next term. If we subtract 6 from 9, the difference
is 3. We see that this difference is the same if we subtract
9 from 12, 12 from 15, 0 from 3, or 3 from 6. Once you determine
the difference between the terms, it is easy to generate the
next terms in the sequence:
0, 3, 6, 9, 12,
15, 18, 21, …
It may also be
helpful for some problems to translate the sequence into a general
form. In this case, we could represent this as n, n+3, (n+3)
+3, … Keep in mind, however, that this is useful only in
thinking about the general trend, that is, what is the nature
of the relationship between the terms. It does not, however,
substitute for the actual sequence, since it gives us no information
as to the starting point. The general equation representation
of n, n+3, (n+3) +3, … could represent 0, 3, 6, 9,…
or it could represent 1, 4, 7, 10,… or it could be 0.5,
3.5, 6.5, 9.5, …, etc. Keeping that caveat in mind, when
solving problems involving more complex sequences, it is often
useful to jot down the general formula describing that sequence.
Let's look at
another example. What are the next two terms of the following
sequence?
2, 4.5, 7, 9.5,
12, 14.5, …
It's usually
easiest to start with the first terms in the series, as those
are typically the smaller numbers, and thus easier to manipulate
quickly. If we subtract 2 from 4.5, we get 2.5. If we subtract
4.5 from 7, the difference is also 2.5. Going through all the
given pairs of terms in the sequence, we can confirm that this
relationship holds true, that each term can be generated through
the addition of 2.5 to the preceding term. (Note: if this relationship
were not to hold true for each pair of terms, we would decide
that this is not an arithmetic progression, and we would then
test whether it were instead a multiplicative progression, described
below.) Once we determine the constant value of 2.5, it is straightforward
to generate the next two terms in the sequence, 17 and 19.5.
In this case, the general form of the sequence would be n, n
+ 2.5,…
Let's try another
sequence. What are the next two terms in the following sequence?
19, 12, 5, -2,
-9, …
Again, to determine
the next two terms, we first must determine the constant value
that is the difference between pairs of terms. If we subtract
19 from 12, we have a difference of -7. If we subtract 12 from
5, again we have -7, and likewise for the remaining terms of
the sequence. Once we know this value, we can quickly ascertain
the next two terms, -16 and -23. How could we write this in a
general form? n, n + (-7),… This example should have also
demonstrated to you another important feature of arithmetic progressions:
the constant values may be either positive or negative. Keep
this in mind when solving these problems.
Multiplicative
Progressions
In a multiplicative
progression, the ratio of consecutive terms is constant. Rather
than adding a constant value to a term in order to generate the
next term as we do for arithmetic progressions, for multiplicative
progressions, we multiply each term by a constant value to generate
the next term. For example, let's consider the following sequence:
1, -3, 9, -27,
81, …
First, let's
determine the constant value separating sets of consecutive terms.
Depending on your personal preference, you can think about it
in terms of division or multiplication. What number must we multiply
1 by in order to get -3 as a product? Alternatively, -3 divided
by what number results in 1? The answer (in both cases) is -3.
Next, confirm that the constant value can be multiplied to each
of the terms in the sequence to generate the next (that is, -3
x -3 = 9; 9 x -3 = -27; and -27 x -3 = 81). In general form,
this would be n, n x (-3),…
Let's look at
another example of a geometric progression:
64, 32, 16, 8,
4, …
What is the next
term in this sequence? First, determine the constant value. 64
multiplied by what number results in a product of 32? Or, what
number must 32 be divided by to result in 64? The answer is 0.5,
or ½. Again, confirm that each term can be multiplied
by this constant value to generate the following term. Once you
have established that you are correct in identifying this as
a geometric progression, you can generate the next term by multiplying
the constant value of 0.5 or ½ by 4, resulting in 2. In
general form: n, n x ½,…
More complex
sequences
Now, not all
sequences will be either arithmetic or multiplicative progressions.
Some sequences will be defined by more complex functions than
either addition or multiplication alone. For example, imagine
a sequence of numbers in which each term (except the first two)
is generated by the addition of the two preceding terms:
1, 1, 2, 3, 5,
8, 13, …
This is actually
a well-studied sequence of numbers called the Fibonacci series.
After the first two numbers, we can represent each term of the
series as n = (n-1) + (n-2), or n is the sum of the two preceding
numbers. (By using subscripts, we are indicating the previous
terms, where n-1 is the term immediately preceding n and
n-2 is the term immediately preceding n-1.)
Alternatively,
you may see a sequence in which each subsequent term is derived
from both arithmetic and multiplicative processes. For example,
the following sequence begins with 2, and all subsequent terms
are generated by adding 1 to the preceding term, and then multiplying
that sum by 2:
2, 6, 14, 30,
62, …
The general representation
of this, not including the first term, would be n = ((n-1) +
1) x 2, or n equals the sum of the preceding term and 1, multiplied
by 2.
For sequences
of this type, it is much more difficult to determine the relationship
between the terms at first glance. However, it is rare that you
would actually be asked to do this on the GMAT . Instead, you
may see a problem in which they define the relationship for you
(as in giving you the rule to add 1 to the preceding term, and
then multiply the sum by 2), and then ask you to generate the
next term of the sequence.
Sample problems
Let's now look
at some sample sequence problems that you might find on the actual
exam.
Example 33
Except for the first two numbers, every number in the sequence
1, -2, -2, 4, … is the product of the two immediately preceding
numbers. What is the seventh term of this sequence?
(A) -8
(B) 32
(C) -32
(D) 256
(E) -256
Answer explanation:
(D) We are given the rule to use in order to generate the next
terms of the sequence: multiply the two immediately preceding
numbers to generate the next. We already have the first four
terms, and need to generate three more. -2 x 4 = -8, this is
the fifth term in the sequence. 4 x -8 = -32, this is the sixth
term. -8 x -32 = 256, this is the seventh term, and the correct
answer, choice D.
Example 34
The fifth term in a sequence of numbers is 19 and each number
after the first number in the sequence is 3 less than the number
immediately preceding it. What is the second number in the sequence?
(A) 31
(B) 30
(C) 28
(D) 13
(E) 10
Answer explanation:
(C) This is an example of an arithmetic progression in which
we are given only one term and asked to determine another. We
are told that each term is 3 less than the previous term. A good
technique to use in solving this is to draw out five blanks to
represent the terms of the sequence, and fill in the fifth one,
the one term we know, with 19:
We are told that
each term is 3 less than the term immediately preceding it. Does
that mean that the fourth term, the one immediately preceding
19, will be 3 more or 3 less? (Be sure to read carefully!) The
fourth term will be three more than 19, or 19 + 3, or 22. In
this way, we can work backwards to generate each of the first
four terms, resulting in a sequence that looks like this:
Now we can see
that the second term is 28, choice C.
Example 35
What is the next term of the sequence -3, 6, -12, 24,…?
(A) -48
(B) 48
(C) -64
(D) 64
(E) -144
Answer explanation:
(A) This is a multiplicative progression generated by multiplying
each term by the constant value of -2 in order to generate the
next term. (-3 x -2 = 6; 6 x -2 = -12, etc.) To generate the
next term, multiply 24 by -2 to get -48, choice A.
Example 36
In a sequence of integers, A, B, C, D, E…, the value of
each integer except the first is equal to two more than the product
of the previous integer and 2. If E equals 14, what is the value
of B?
(A) -14
(B) -8
(C) 0
(D) 4
(E) 8
Answer explanation:
(C) This is one of the more complex sequences in which each subsequent
term is derived from both arithmetic and multiplicative processes.
Like that previous problems, it may be useful to create a little
diagram to hold the places for the terms as you figure them out.
We are told that the fifth term, E, equals 14, so we can fill
in that information as follows:
We are told that
the value of each integer is equal to two more than the product
of the previous integer and 2. A general representation of this
would be n = ((n-1) x 2) + 2. It is extremely important to be
very precise in your interpretation of the description of the
relationship between the terms of the sequence. In this case,
a number is multiplied by 2, then 2 is added to that product
in order to generate the next term.
Now, if we use
this general formula and substitute our known term, 14, for n,
we can derive the preceding term as follows:
n = ((n-1) x
2) + 2
14 = ((n-1) x
2) + 2
Subtracting 2
from both sides, we get
12 = ((n-1) x
2)
Dividing both
sides of the equation by 2, we get
6 = n-1
Therefore, position
D is 6. Knowing the value of D, we can now apply the formula
again to solve for C. Once we have C, we can solve for B. Finally,
we end up with the sequence as follows:
Now we can see
that B = 0, choice C.
Summary for sequence
problems:
- If you are asked
to solve for a term in the sequence, determine what the relationship
is between the terms of the sequence.
- Is the sequence
an arithmetic progression, multiplicative progression, or a more
complex sequence defined within the problem? If you think it's
an arithmetic progression, determine the common value and make
sure that each term of the sequence can be generated through
the addition of that common value to the preceding term. Likewise,
if you think it's a multiplicative progression, determine the
common value and confirm that each term can be multiplied by
this constant value to generate the following term.
- If it helps,
write out a more general representation of the sequence or the
formula used to determine the next term of a sequence using n
to represent any given term.
- When you are
asked to solve for the value of a specific term, it may be useful
to make a little diagram to ensure that you are solving for the
correct position.
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