ACT and SAT Math Tips: Sequences

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There are three main typesNumbers-1.jpg of sequence questions on the ACT and SAT:

  1. Those that require the use of formulas to solve arithmetic or geometric sequences.
  2. Those that ask you to compute a small-numbered term (such as the 8th term or less).
  3. Those that assess your ability to discover a repetitive pattern in order to find a higher-numbered term (such as the 51st term).

It is this third type of sequence--often considered the most difficult by unprepared test takers--that we will address today. Like many ACT and SAT math questions, there is a trick that makes these sequence questions quite easy to solve.These questions typically ask you to find a two-digit or even a three-digit term. Let's look at an example:

In the sequence 2, –6, 8, ..., the first term is 2 and the first three terms repeat continuously. What is the 41st term in the sequence?

You could write out all 41 numbers, but this solution method is time-consuming and inefficient. A better method is to establish a pattern.

In this example, calculations are not required. The pattern is simply 2, –6, 8, 2, –6, 8, 2, ...
        Sequence1.png

There are three numbers in the pattern. Therefore, all terms that are multiples of three are 8. The 3rd term, the 6th term, and the 9th term are all 8. The 12th term, 15th term, 18th term, etc. are also all 8. What multiple of 3 is close to 41?
   Sequence2.png

3 × 13 = 39    The 39th term is 8, the 40th term is 2, and the 41st term is –6. 

More difficult questions may require you to compute the first six to eight terms in order to establish the repetitive pattern:

Sequence3.pngCalculate the terms of the sequence until you find a pattern:
                        Sequence4.png
The pattern repeats after every 4 numbers. Therefore, multiples of 4 will help us find our answer. All terms that are multiples of 4 have a value of 0. What multiple of 4 is close to 45? You can use 4 × 10 = 40 or 4 × 11 = 44:
                                               
Sequence5.png
The correct answer is (B).  

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Image: "Numbers" courtesy of morebyless