GMAT Quantitative: Sequences and Series

Mastering sequence and series problems on the GMAT is less about rote memorization and more about developing a keen eye for pattern recognition and systematic algebraic thinking. These questions test your ability to discern order from apparent chaos, a skill directly applicable to the logical data analysis required in business. Success here comes from understanding a core set of definitions and applying a structured, step-by-step approach to unlock any sequence you encounter.

Foundational Definitions and The Systematic Approach

A sequence is an ordered list of numbers, and each number in the list is called a term. A series is the sum of the terms of a sequence. The most critical first step in any GMAT sequence problem is to identify the type of pattern governing the progression. Jumping straight into calculations without this diagnosis is a common source of errors.

Your systematic approach should always begin with three actions: 1) Write down the first few given terms. 2) Calculate the differences between consecutive terms (for an additive pattern). 3) Calculate the ratios between consecutive terms (for a multiplicative pattern). This initial investigation will almost always point you toward the correct formula and solution path.

Arithmetic Sequences: The Constant Additive Pattern

An arithmetic sequence is defined by a constant difference between consecutive terms. This constant is called the common difference, denoted by $d$. If the first term is $a_1$, the sequence progresses as $a_1,a_1+d,a_1+2d,a_1+3d,...$

The formula for the $n$th term ($a_n$) is: $$a_n=a_1+(n-1)d$$

The sum of the first $n$ terms ($S_n$) can be found by averaging the first and last term and multiplying by the number of terms: $$S_n=\frac{n(a_1+a_n)}{2}$$

GMAT Example: In a sequence, the 4th term is 10 and the 10th term is 34. What is the 16th term? Step 1: Use the $n$th term formula to create equations. $a_4=a_1+3d=10$ $a_{10}=a_1+9d=34$ Step 2: Subtract the first equation from the second to solve for $d$. $(a_1+9d)-(a_1+3d)=34-10$ → $6d=24$ → $d=4$ Step 3: Plug $d=4$ into the first equation to find $a_1$. $a_1+3(4)=10$ → $a_1=-2$ Step 4: Solve for the 16th term. $a_{16}=-2+(15)(4)=-2+60=58$

Geometric Sequences: The Constant Multiplicative Pattern

A geometric sequence is defined by a constant ratio between consecutive terms. This constant is called the common ratio, denoted by $r$. The sequence progresses as $a_1,a_{1}r,a_1{r}^2,a_1{r}^3,...$

The formula for the $n$th term is: $$a_n=a_1\cdot r^{(n-1)}$$

The sum of the first $n$ terms is: $$S_n=a_1\cdot \frac{1-r^n}{1-r}\text{(for }r\ne 1\text{)}$$

On the GMAT, you will also encounter the concept of an infinite geometric series. Such a series has a sum only if the common ratio $r$ is between -1 and 1 (i.e., $|r|<1$). In this case, the sum converges to a finite value given by: $$S_{\infty }=\frac{a_1}{1-r}$$

GMAT Application: A ball is dropped from a height of 64 feet. On each bounce, it rebounds to 3/4 of its previous height. What is the total vertical distance the ball travels until it comes to rest? (Hint: It travels down and up for every bounce after the initial drop). Solution: The down-only distances form a geometric sequence: 64, 48, 36,... with $a_1=64$ and $r=3/4$. The total down distance is an infinite sum: $S_{\infty }=64/(1-3/4)=256$ feet. The up distances are the same sequence but start at 48: $48/(1-3/4)=192$ feet. Total distance = Initial drop (64) + Total up (192) + Total subsequent down (256 - 64) = 64 + 192 + 192 = 448 feet.

Recursive Sequences and Complex Patterns

Some GMAT sequences are defined recursively, meaning each term is defined based on the preceding term(s). For example, $a_1=2$, and $a_n=2a_{n-1}+1$ for $n>1$. The key is to compute terms methodically until you see an emergent pattern or can relate the term you need back to the first term.

Other problems involve repeating patterns or digit patterns. For instance, the sequence of units digits of powers of 7 (7, 9, 3, 1, 7, 9,...) repeats every 4 terms. To find the 47th term in such a repeating cycle, find the remainder when the term number is divided by the cycle length. $47÷4$ gives a remainder of 3, so the 47th term is the 3rd element in the cycle (which is 3).

Common Pitfalls

1. Misidentifying the Sequence Type: Assuming a sequence is arithmetic when it is geometric (or vice versa) because the first few differences look close. Correction: Always check both the difference and the ratio for the first few terms to be certain.
2. Incorrect Indexing in Formulas: Plugging the term number ($n$) directly into the formula without adjusting for $(n-1)$. For the 10th term, you use $n=10$, but the exponent in the geometric formula is $9$, and the multiplier in the arithmetic formula is $9d$. Correction: Write the formula explicitly and substitute carefully.
3. Misapplying the Infinite Sum Formula: Using $S_{\infty }=a_1/(1-r)$ for a geometric sequence where $|r|\ge 1$. This formula only works for convergent series. Correction: Verify the absolute value of $r$ is less than 1 before applying the infinite sum formula.
4. Overcomplicating Recursive Sequences: Trying to find a direct $n$th term formula immediately. Correction: Calculate the first 3-5 terms by hand. A pattern (often arithmetic or geometric in disguise) usually reveals itself, allowing you to solve without deriving a complex closed form.

Summary

• The cornerstone of sequence problems is pattern identification. Systematically test for a common difference (arithmetic) or a common ratio (geometric) first.
• Know the core formulas cold: $a_n=a_1+(n-1)d$ and $S_n=\frac{n(a_1+a_n)}{2}$ for arithmetic sequences; $a_n=a_1\cdot r^{(n-1)}$ and $S_n=a_1\cdot \frac{1-r^n}{1-r}$ for geometric sequences.
• For an infinite geometric series, a finite sum exists only if $|r|<1$, calculated with $S_{\infty }=\frac{a_1}{1-r}$.
• Tackle recursive and repeating pattern problems by writing out terms methodically. For repeating cycles, use the remainder of the term number divided by the cycle length to find your position.
• Avoid indexing errors and formula misapplication by double-checking your sequence diagnosis and your substitution into formulas. On the GMAT, sequence questions are designed to be solved efficiently with the right approach.