🔢 Quant

Sequences, Series & Estimation

Arithmetic and geometric sequences, recursive rules and cycles, plus disciplined estimation.

~4h

to master

Sequences: rules, not lists

A sequence is a function on positions $1, 2, 3, \dots$ — given either explicitly ( $a_n = 3n + 2$ ) or recursively ( $a_n = 2a_{n-1} + 1$ with a start value). The GMAT tests whether you can run the rule forward, jump to a distant term, or spot a repeating pattern instead of grinding.

Arithmetic sequences (constant difference $d$ )

📐Core Rule

$a_n = a_1 + (n-1)d$ . The 50th term of $5, 8, 11, \dots$ is $5 + 49 \cdot 3 = 152$ . The $(n-1)$ is the fence-post again — 49 steps reach the 50th post.

Sum of $n$ terms: $\text{sum} = n \cdot \dfrac{a_1 + a_n}{2}$ — count × average of the ends, because evenly spaced lists have mean = midpoint. The sum $1 + 2 + \cdots + 100 = 100 \cdot \frac{101}{2} = 5050$ .
Consecutive integers/evens/odds are arithmetic with $d = 1$ or $2$ — every "consecutive" question is this machinery.

Jump to a far term on .

Geometric sequences (constant ratio $r$ )

📐Core Rule

$a_n = a_1 \cdot r^{\,n-1}$ . Doubling is $r = 2$ ; a 5% annual increase is $r = 1.05$ — compound interest is a geometric sequence wearing a suit.

Growth questions ("after how many steps does it first exceed…") are usually fastest by direct iteration with small numbers: double 3 → 6, 12, 24, 48, 96, 192… and count steps as you go. Try .

💡Exam Tip

Messy recursions often telescope or cycle. Compute the first 4–6 terms of any unfamiliar rule. Sequences like $a_n = \frac{1}{1 - a_{n-1}}$ repeat with period 3 — so $a_{100}$ is just $a_{100 \bmod 3}$ -ish bookkeeping, not 99 iterations. If the GMAT asks for term 100, there is always a shortcut.

Estimation: the licensed shortcut

When a question says "approximately" — or when answer choices are far apart — exact arithmetic is wasted time.

Round to friendly anchors: $\frac{4.02 \times 29.8}{0.61} \approx \frac{4 \times 30}{0.6} = 200$ .
Bound it: $\sqrt{52}$ sits between $\sqrt{49}=7$ and $\sqrt{64}=8$ , nearer 7. Squeezing between known squares/cubes answers most root questions.
Compare to benchmarks: is $\frac{7}{15}$ more or less than $\frac{1}{2}$ ? Compare $7 \cdot 2$ vs $15$ . Benchmark fractions ( $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ ) settle comparisons without common denominators.

⚡Shortcut

Check the spread first. Options $0.3, 3, 30, 300$ ? One significant figure of work decides it. Options $268, 274, 281$ ? Estimation can only eliminate, not decide — switch to exact mode. The choices tell you how much precision you owe.

⚠️GMAT Trap

Round late, and in one direction deliberately. Rounding both 4.02 → 4 and 0.61 → 0.6 mid-chain can drift the result past a neighbouring option when choices are close. When you must round, know whether each rounding pushed your estimate up or down — that tells you which neighbouring option the true value sits toward.

Checklist

$(n-1)$ steps to the $n$ th term — both formulas
Arithmetic sum = count × average of ends
Unknown recursion → compute terms, hunt the cycle
Far-apart options → estimate; tight options → exact
Track the direction of every rounding

Sample Questions

23 practice questions

Medium

A state produced 840 million pounds of milk in 2011 and 2.2 billion pounds in 2016. Approximately how many more million gallons of milk did it produce in 2016 than in 2011? (1 billion = $10^9$ , and 1 gallon of milk weighs 8.6 pounds.)

Medium

Let $M = \frac{1}{101} + \frac{1}{102} + \frac{1}{103} + \cdots + \frac{1}{200}$ , the sum of the reciprocals of every integer from 101 through 200. Which of the following must be true?

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