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Sequences on the GMAT

Understand how to handle these tricky upper level Quant problems! 

Definitions

A sequence is a list of numbers that follow some mathematical patterns.  More formally, a sequence is a function whose inputs are limited to the positive integers.  Terms are denoted by a letter for the whole sequence, and in the subscript, the index, which is the place on the list.  Here are some examples of common sequences.  First, (A) and (B) are arithmetic sequences:

A) a_1= 4, a_2= 7, a_3= 10, a_4= 13, a_5= 16, …

B) t_1= 9, t_2= 5, t_3= 1, t_4= -3, t_5= -7, …

In an arithmetic sequence, the same “common difference” is added or subtracted each time.  In (A), we add 3 to each term to get the next.  In (B), we subtract 4 to get each new term.  If you graphed the terms of an arithmetic sequence against the indices, the dots would follow a straight line in the x-y plane.

The next two, sequences (C) and (D), are geometric sequences:

C) e_1= 2, e_2= 6, e_3= 18, e_4= 54, e_5= 162, …

D) y_0= 56, y_1= 28, y_2= 14, y_3= 7, y_4= 3.5, …

In a geometric sequence, the same “common ratio” is multiplied or divided each time.  In (C), we multiply each term by 3 to get the next.  In (D), we divide by 2 to get each new term.  If you graphed the terms of an geometric sequence against the indices, the dots would follow an exponential function in the x-y plane.

Notice also — (D) begins with a “zeroth” term.  The first term of a sequence can have an index of either 1 or 0: both appear on the GMAT.

There are other exotic sequences in mathematics.  Here are two more

E)j_1= 1, j_2= 1, j_3= 2, j_4= 3, j_5= 5, j_6= 8j_7= 13j_8= 21, …

F)p_0= 1, p_1= 1, p_2= 2, p_3= 3, p_4= 5, p_5= 7p_6= 11p_7= 15,  …

Both of these are famous in mathematics.  The first is the Fibonacci sequence, which I will discuss further below.  The second, known in math as the Partition Function (http://en.wikipedia.org/wiki/Partition_(number_theory)) gets into way way more difficult math that you need to know for the GMAT.

 

Explicit Series

A explicit series is a series in which the general rule for finding each term can be stated, either verbally or mathematically.  Sometimes the GMAT will give you the general rule for a sequence in algebraic form:

  1. 4
  2. 6
  3. 8
  4. 12
  5. 18

If the rule is given algebraically, all we have to do is plug in the index number to find the value of each term.  Here, we plug in n = 3 to find the third term, and plug in n = 4 to find the fourth term.  The third and fourth terms are:

a_3=(3)^2 - 3 = 9 - 3 = 6

a_4=(4)^2 - 4 = 16 - 4 = 12

so the difference between them is 12 – 6 = 6, which is answer B.  I will give another practice question of this genre at the end.

The explicit description might also be given verbal.  For example

2) In each term of a sequence, 9 is added to get the next term.  If the first term is 2, what is the eighty-first term?

  1. 632
  2. 695
  3. 713
  4. 722
  5. 731

Notice, the rule for the whole sequence is articulated: we know it is an arithmetic sequence with a common difference of 9.  You can try this question now.  I will show the solution at the end of this article.

 

Recursive Sequences

In mathematics, a recursive definition is one where you need the result at end of each step in order to proceed to the next step.  A sequence has a recursive definition if the only way to calculate the third term is first to calculate the second term, and the only way to calculate the fourth term is first to calculate the third term, etc. etc.  Clearly, if we were asked for, say, the sixth term of a recursive sequence, we would have to calculate each and every term along the way: there’s no “shortcut” we can use to shoot directly to the sixth term.

Recursive defined sequence on the GMAT are almost always given in algebraic form.  Here’s an example of a question involving one.

  1. 4
  2. 11
  3. 20
  4. 31
  5. 116

Whenever you see the nth term equal to some expression involving an (n – 1)th term, that’s a recursively defined sequence.  The idea is: to find the n = 3 term, you would have to plug the n – 1 = 2 term into the formula; to find the n = 4 term, you would have to plug the n – 1 = 3 term into the formula; etc.  I will discuss a full solution to this one below.

Sometimes, a sequence is recursively defined not simply in terms of the previous term, but in terms of the previous two terms.  The most famous sequence defined this way is the Fibonacci Sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

This is famous because these numbers are found all over Nature: the number of seeds in almost any fruit, the number of petals on any flower, even the number of digits on one human hand or foot.  Some theorists feel Fibonacci Numbers even give insight into the fluctuations of the stock market: http://www.goldennumber.net/fibonacci-stock-market-analysis/

The Fibonacci series has a recursive definition that can be written algebraically as:

F_n=F_{n-1} + F_{n-2}

Think about what that says.  If I want the n = 5 term, for example, I have to take the sum of the n – 1 = 4 term with the n – 2 = 3 term.  To find any term, I have to add the two previous terms.  You start with two 1’s, follow that rule, and you generate the Fibonacci sequence.

Whenever you see the nth term expressed as some combination of the (n-1)th and (n-2)th terms, that’s a recursively defined sequence in which each term is defined in terms of the previous two terms.  You will always be given two “starter” terms to such a sequence.  Here’s a sample problem involving one.

  1. 1
  2. 7
  3. 22
  4. 155
  5. 721

I will give a complete solution to this below.

If you understand how to plug into the algebraic sequences, and how to extend the patterns and find terms of the verbal defined sequences, you will be able to handle everything the GMAT throws at you involving sequence.

 

One Further Practice Question

As promised, another question of the first type discussed.

  1. 1/2
  2. 2/7
  3. 5/8
  4. 7/10
  5. 7/12

 

Practice Question Explanations

Practice question #1 was explained in the text of the article, immediately following the question, and the answer (B) was stated there.

2) Let’s think about the first few terms of this sequence.

a_1=2
a_2=11 = 2 + 9
a_3=20 = 2 + 9 + 9
a_4=29 = 2 + 9 + 9 + 9

To find each new term, we add 9. The result is, each term equals 2 plus a sum of 9’s.  How many nines?  Well, the second term has one 9, the third term has two 9’s, the fourth term has three 9’s, so it’s clear the nth term would have a sum of (n – 1) 9’s.  Thus, the eighty-first term would be 2 plus eighty 9’s.  Well, eighty 9’s = 80*9 = 720, so the eighty-first term = 722, answer D.

3) This is a recursively defined sequence, so we have to find the values of each term up until the one we want, the 5th term.  Fortunately, the “starter” term we are given is the 2nd term, so we have a bit of a head start.  We don’t know why the question started us with the second term, instead of the first, term, but it works to our advantage, so we won’t complain.  To find the next term, use the recursion formula:

a_n=(a_n -1)^2 - 5
a_3=(a_2 )^2 - 5 = (3)^2 - 5 = 9 - 5 = 4
a_4=(a_3 )^2 - 5 = (4)^2 - 5 = 16 - 5 = 11
a_5=(a_4 )^2 - 5 = (11)^2 - 5 = 121 - 5 = 116

Notice, there’s no shortcut with recursive sequences: we have to find the third term = 4 and then use that to find the fourth term = 11, then used that to find the fifth term = 116.  There’s no way to go directly to the fifth term with calculating each one of the previous terms.  The fifth term = 116, answer E.

4) This is a recursively defined sequence, defined in terms of the previous two terms, so we have to find the values of each term up until the one we want, the 6th term.  Again, we have to find the terms one by one, using the recursion formula.

a_1=1
a_2=1
a_n=(a_n -1) x (a_n - 2) + 1
a_3=(a_2) x (a_1) + 1 = (1)(1) + 1 = 1 + 1 = 2
a_4=(a_3) x (a_2) + 1 = (2)(1) + 1 = 2 + 1 = 3
a_5=(a_4) x (a_3) + 1 = (3)(2) + 1 = 6 + 1 = 7
a_6=(a_5) x (a_4) + 1 = (7)(3) + 1 = 21 + 1 = 22

So, the sixth term = 22, answer = C.

5) Ah, an explicitly defined sequence!  After recursive sequences, this is much easier.  All we have to do to find the 7th term is to plug n = 7 into the formula.

a_n = {n-2}/{n+3}
a_7 = {7-2}/{7+3} = 5/10 = 1/2

The seventh term is 1/2, Answer = A.

 

Even More Practice Questions

6) http://gmat.magoosh.com/questions/925

7) http://gmat.magoosh.com/questions/807

 

About the Author

Mike McGarry is a Content Developer for Magoosh with over 20 years of teaching experience and a BS in Physics and an MA in Religion, both from Harvard. He enjoys hitting foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Follow him on Google+!

12 Responses to Sequences on the GMAT

  1. Julian November 7, 2014 at 7:59 am #

    Hey,

    small error in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 35, 56, 89, 144

    instead of 1,1,2,3,5,8,13,21,34,55.

    Cheers, Julian

    • Mike
      Mike November 7, 2014 at 11:45 am #

      Dear Julian,
      Ooops! That was a rather substantial mistake on my part! I should have known better with such famous mathematical example! Thank you very much for pointing this out! :-) I just fixed that error in the blog. You have a very good idea for detail, and that will serve you well on the test. Thank you very much for pointing this out! Best of luck to you, my friend. :-)
      Mike :-)

  2. Hets September 23, 2014 at 1:10 am #

    Hello,

    Question 3 says:
    a_n=(a_n -1)^2 – 5
    a_3=(a_2 )^2 – 5 = (3)^2 – 5 = 9 – 5 = 4

    Shouldn’t a_3= (a_2 -1)^2-5??
    Thanks.
    ======================================================
    Sorry I realised my mistake! The solution is right

    • Mike
      Mike September 23, 2014 at 10:14 am #

      Dear Hets,
      I’m happy to respond :-) and I am very glad for you that you figured out your own mistake. That, of course, is the very best kind of learning, when you can figure out your own mistakes and move past them. I wish you as much of this kind of learning as possible! :-)
      I hope that you don’t mind that I posted your question & retraction, because I thought the issue would be helpful for other students to read as well. Mathematics, after all, is about precision —- precision down to the smallest detail. A careful eye for detail is absolutely essential for a truly excellent Quant performance on the GMAT.
      Does all this make sense?
      Mike :-)

      • Hets September 24, 2014 at 12:48 am #

        Hi Mike,

        No worries, I think my brain had stopped working after 6 hours of studying! Thank you so much for your prompt reply, and also would like to thank you for your amazing work, it is really helping me with my GMAT preparation!

        I had another question regarding absolute values, a DS question: If xy does not equal 0, is x > y?
        Statement 1: 4x=3y
        Statement 2: Iy-xI = x-y

        For statement 2, the explanation was :
        X Y x>y
        3 3 No
        -3 -4 yes

        I understand how y=3 when x=3, but do not understand how y=-4, when x=-3.
        Can you please explain?

        (I wanted to post this on the page with this question, but can’t seem to find it!)

        Thanks!

        • Mike
          Mike September 24, 2014 at 10:13 am #

          Dear Het,
          First of all, you are quite welcome. :-) Thank you for your kind words.
          Now, for this question, it’s a Magoosh question you can see here:
          http://gmat.magoosh.com/questions/966
          Watch the VE carefully, and read the TE carefully, and I think that will clear up your questions.
          Mike :-)

          • Hets September 29, 2014 at 1:26 am #

            Hi Mike,

            Yes I get it now! It won’t work if y > x!

            Thanks Mike!

            • Mike
              Mike September 29, 2014 at 10:04 am #

              Dear Hets,
              You are quite welcome. :-) I am very glad you found that helpful. Best of luck to you in the future, my friend.
              Mike :-)

  3. Perhaps April 6, 2013 at 2:50 am #

    hi magoosh,
    i cant find the solution to the question 4 ….where is it???

    • Mike
      Mike April 6, 2013 at 11:08 am #

      Dear Perhaps,
      Under the section “Practice Question Explanations”, you will see the solutions to #2, then #3, then #4, and then #5. The answer to #4 is (C), and it’s explained right there.
      Does this make sense?
      Mike :-)

  4. utsab January 31, 2013 at 7:50 am #

    At the top of the page, in the example B for arithematic sequences, the given sequence shows: T1=9,T2=5,T3=1,T4=-7,T5=-11 . Passage below the example states that this is a squence involving difference of 4. So, shouldn’t T4= -2 and T5=-7?

    • Mike
      Mike January 31, 2013 at 9:57 am #

      Dear Utsab,
      Yes, you are perfectly correct. That was a typo. I just fixed it. Thank you for pointing that out, Best of luck to you!
      Mike :-)


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