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GMAT Exam: A Simple Guide To Solving Word Problems - Sequences

A sequence is a list of terms written in a particular order.
In an arithmetic sequence, consecutive terms have a common difference.

The sequence has the form a1, a1 + d, a1 + 2d, …, a1 + (n – 1)d, …, where a1 is the first term and d is the common difference between terms.

The general term for an arithmetic sequence is an = a1 + (n – 1)d.

For example, the first five terms of the arithmetic sequence that has first term a1 = 3 and common difference d = 5 are 3, 8, 13, 18, and 23.

The 20th term is a20 = 3 + (20 – 1)(5) = 3 + (19)(5) = 98.

In a geometric sequence, consecutive terms have a common ratio.

The sequence has the form a1, a1r, a1r2, …, a1rn–1, …, where a1 is the first term and r is the common ratio.

The general term for a geometric sequence is an = a1rn–1.

 

For example, the first five terms of the geometric sequence that has first term a1 = 10 and common ratio r = 2 are 10, 20, 40, 80, and 160.

The 10th term is a10 = (10)(210–1) = (10)(29) = (10)(512) = 5,120.

 

A recursive definition for a sequence includes the value of one or more initial terms and a formula for finding each term from previous terms.

The Fibonacci sequence has the following recursive definition: a1 = 1, a2 = 1, and an = an–1 + an–2, for n ≥ 3.

The first five terms of the sequence are 1, 1, 2, 3, and 5.

 

If you encounter sequences on the GMAT, look for a recognizable pattern.

Is there a common difference? Is there a common ratio? Are the terms sequential perfect squares or cubes? Are the terms obtained by adding or subtracting the previous terms in some way?

Example: The first five terms of a sequence are 2, 5, 8, 11, and 14. If the sequence continues in the same manner, what is its 50th term?
The terms shown have a common difference of 3 with first term a1 = 2. If this pattern continues in the same manner, the general term is a1 + (n – 1)d = 2 + (n – 1)(3). Thus, the 50th term is a50 = 2 + (50 – 1)(3) = 2 + (49)(3) = 149.