QUANTITATIVE REASONING · GMAT-Q-SEQ

Sequences and Patterns

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

a_{n} = a_{1} + (n - 1) d

Translate the condition

An arithmetic sequence changes by the same amount at each step. From term one to term n there are n minus one equal steps.

Test the result

Verify the term number before counting steps. The eighth term is seven common differences after the first term.

Think Like a GMAT Test-Taker

Identify the term number, starting value, and constant step before extrapolating.

Faster Methods

Translate the target into an equation before calculating.

Check the result against the original conditions and units.

Worked example 1

An arithmetic sequence begins with 7 and has common difference 2. What is its eighth term?

a_{8} = 7 + (8 - 1) 2 = 21

From the first term to the eighth term there are seven equal steps; add 7 times the common difference 2 to 7 to obtain 21.

Worked example 2

An arithmetic sequence has fifth term 19 and ninth term 35. Find its first term.

d = \frac{35 - 19}{9 - 5} = 4, a_{1} = 19 - 4 (4) = 3

Count the gaps between term positions. There are four gaps from the fifth to the ninth and four gaps from the first to the fifth. Subtract four common differences to recover the first term.

Worked example 3

A sequence begins with 5 and each later term is twice the previous term minus 1. Find its fourth term.

a_{1} = 5, a_{2} = 2 (5) - 1 = 9, a_{3} = 17, a_{4} = 33

Make a short index-and-value list. Apply the full recursive rule once per step. This is not an arithmetic sequence because its consecutive differences are not constant.

Common Traps

Keep domain restrictions after simplifying.
Do not confuse the requested output with an intermediate quantity.