🔢 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

22 practice questions

Medium

In a theater, the first row has 14 seats, and each subsequent row has 3 more seats than the row in front of it. If the theater has 25 rows, how many seats are in the last row?