Sequences

Sequences are ordered lists of numbers that follow a specific pattern or rule. Understanding sequences is crucial for solving many GMAT problems, especially in the Quantitative section. Sequences can be arithmetic, geometric, or follow other patterns.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference, denoted by $d$.

The general form of an arithmetic sequence is:

$a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \dots$

where:

• $a_1$ is the first term,
• $d$ is the common difference.

The $n^{th}$ term of an arithmetic sequence can be calculated using the formula:

$a_n = a_1 + (n - 1)d$

The sum of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n}{2} \times (a_1 + a_n)$

Geometric Sequences

A geometric sequence is a sequence of numbers where the ratio between consecutive terms is constant. This ratio is called the common ratio, denoted by $r$.

The general form of a geometric sequence is:

$a_1, a_1 \times r, a_1 \times r^2, a_1 \times r^3, \dots$

where:

• $a_1$ is the first term,
• $r$ is the common ratio.

The $n^{th}$ term of a geometric sequence can be calculated using the formula:

$a_n = a_1 \times r^{n-1}$

The sum of the first $n$ terms of a geometric sequence is given by:

$S_n = a_1 \times \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)

Special Sequences

Some sequences do not follow arithmetic or geometric patterns but have unique rules. Examples include:

• Fibonacci Sequence: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8, ...).
• Square Numbers: Each term is the square of an integer (e.g., 1, 4, 9, 16, ...).
• Cube Numbers: Each term is the cube of an integer (e.g., 1, 8, 27, 64, ...).

Applications in GMAT Problems

Sequences are often tested in GMAT problems involving:

• Finding specific terms in a sequence.
• Calculating the sum of a sequence.
• Identifying the pattern or rule governing a sequence.
• Solving word problems involving sequences.

Tips for Solving Sequence Problems

• Identify whether the sequence is arithmetic, geometric, or follows another pattern.
• Use the appropriate formula to find the $n^{th}$ term or the sum of terms.
• Look for patterns or relationships between terms.
• Practice identifying sequences quickly to save time on the exam.

Mastering sequences will help you efficiently solve a variety of GMAT problems and improve your overall quantitative score.