Sequences: complete GMAT guide

Sequences: complete GMAT guide | GMAT Sprint

Sequence questions ask you to find a term, a sum, or a pattern. The GMAT focuses on arithmetic sequences, where the difference between terms is constant, and geometric sequences, where the ratio between terms is constant. Some questions define a sequence by a rule instead of a formula, and those require careful substitution.

Arithmetic sequences

An arithmetic sequence increases or decreases by a constant difference $d$ . If the first term is $a_1$ , the $n$ th term is

a_n=a_1+(n-1)d

For example, the sequence 3, 7, 11, 15, ... has $a_1=3$ and $d=4$ . The 10th term is $3+(10-1)4=39$ .

The sum of the first $n$ terms is the average of the first and last terms multiplied by the number of terms:

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

For the same sequence, the sum of the first 10 terms is $\frac{10(3+39)}{2}=210$ .

Example. What is the sum of the even integers from 2 to 100 inclusive?

This is an arithmetic sequence with $a_1=2$ , $a_n=100$ , and $d=2$ . The number of terms is $(100-2)/2+1=50$ . Sum: $\frac{50(2+100)}{2}=2550$ .

Finding the number of terms

A common error is counting the terms incorrectly. The number of terms in an arithmetic sequence is

n=\frac{a_n-a_1}{d}+1

The plus one matters. From 5 to 25 with a difference of 5, the terms are 5, 10, 15, 20, 25, so there are 5 terms. The formula gives $(25-5)/5+1=5$ .

Geometric sequences

A geometric sequence multiplies by a constant ratio $r$ . The $n$ th term is

a_n=a_1 r^{n-1}

For example, 2, 6, 18, 54, ... has $a_1=2$ and $r=3$ . The 5th term is $2 \cdot 3^4=162$ .

The sum of the first $n$ terms is

S_n=a_1\frac{1-r^n}{1-r}

for $r \neq 1$ . If $r=1$ , the sum is simply $na_1$ .

Example. A ball is dropped from a height of 8 feet and bounces to half its previous height each time. What is the total distance traveled when it hits the ground for the fourth time?

The ball falls 8 feet, then rises 4 and falls 4, then rises 2 and falls 2, then rises 1 and falls 1. Total distance: $8+2(4)+2(2)+2(1)=8+8+4+2=22$ feet.

Recursive sequences

A recursive rule defines each term using previous terms. For example, $a_1=2$ and $a_{n+1}=3a_n-1$ . The first few terms are 2, 5, 14, 41, ... To find a later term, compute the terms in order or look for a pattern.

Some recursive sequences repeat. If a rule produces a cycle, identify the cycle length and use the remainder of the term number divided by the cycle length. For example, if a sequence repeats every 4 terms, the 30th term is the same as the 2nd term because $30=4 \times 7+2$ .

Patterns and differences

When a sequence does not fit an arithmetic or geometric pattern, look at the differences between consecutive terms. If the first differences are constant, the sequence is arithmetic. If the second differences are constant, the sequence is quadratic, and the terms can be described by a quadratic expression.

Example. Find the next term: 2, 6, 12, 20, 30, ...

First differences: 4, 6, 8, 10. The differences increase by 2, so the next difference is 12. The next term is $30+12=42$ .

Sum formulas and pairing

For an arithmetic sequence, pair the first and last terms, the second and second-to-last, and so on. Each pair has the same sum. This is why the sum formula uses the average of the first and last terms.

The sum of the first $n$ positive integers is

1+2+3+\cdots+n=\frac{n(n+1)}{2}

The sum of the first $n$ odd integers is $n^2$ . These formulas appear in questions about consecutive integers and patterns.

Sequences with variables

Questions may define a sequence in terms of a variable, such as $a_n=2n+3$ . To find a specific term, substitute the term number. To find which term equals a given value, set the formula equal to that value and solve for $n$ .

If the sequence is arithmetic with a variable first term, use the general formulas and solve for the unknown. For example, if the 5th term is 17 and the 9th term is 33, the difference is $(33-17)/(9-5)=4$ , and the first term is $17-4(4)=1$ .

Alternating and repeating patterns

Some sequences alternate between two values or follow a cycle longer than two terms. If a sequence is 4, 9, 4, 9, 4, ..., the pattern repeats every two terms. The odd-numbered terms are 4 and the even-numbered terms are 9. For a term number $n$ , divide by the cycle length and use the remainder.

Other sequences combine arithmetic and geometric behavior. For example, a sequence might add 3, then double, then add 3, then double. To find a later term, write out the first several terms until the pattern is clear, or express the rule in two steps. Do not assume a single formula if the rule changes.

Example. A sequence starts with 1. Each term is obtained by multiplying the previous term by 2 and adding 1. What are the first four terms?

$a_1=1$ . $a_2=2(1)+1=3$ . $a_3=2(3)+1=7$ . $a_4=2(7)+1=15$ . The terms are 1, 3, 7, 15.

Sequences and averages

In an arithmetic sequence, the average of all terms equals the average of the first and last terms. This is why the sum formula multiplies the average by the number of terms. For the sequence 5, 8, 11, 14, 17, the average is $(5+17)/2=11$ , which is also the middle term.

For an even number of terms, the average is the average of the two middle terms. The sequence 5, 8, 11, 14 has an average of $(8+11)/2=9.5$ , which is also $(5+14)/2=9.5$ . This relationship can answer questions about sums without listing every term.

If a sequence question gives a sum and asks for a term, try using the average. In an arithmetic sequence, the average is often the middle term or the average of the two middle terms. This shortcut avoids solving for the first term and the difference separately.

When a sequence is defined by a formula with a variable, check whether the formula produces integers. If the sequence is supposed to list counts or positions, a fractional term signals an error. Substitute the term number and simplify before deciding that the pattern is wrong.

A repeatable approach

Check whether the sequence is arithmetic, geometric, or recursive.
Find the first term and the common difference or ratio.
Use the nth-term formula for a specific term.
Use the sum formula for a total.
For patterns, list differences until a rule appears.
Check the number of terms, especially when counting inclusively.

Common traps

Using $n$ instead of $n-1$ in the nth-term formula.
Forgetting the plus one when counting terms.
Averaging terms when the sequence is not arithmetic.
Confusing the common difference with the common ratio.
Assuming a pattern continues without checking more terms.
Missing the first term in a sum.

Quick reference

Concept	Formula
Arithmetic nth term	$a_1+(n-1)d$
Arithmetic sum	$n(a_1+a_n)/2$
Number of terms	$(a_n-a_1)/d+1$
Geometric nth term	$a_1r^{n-1}$
Geometric sum	$a_1(1-r^n)/(1-r)$
Sum of first $n$ integers	$n(n+1)/2$
Sum of first $n$ odds	$n^2$

Sequence questions are pattern questions. Identify whether the pattern is additive or multiplicative, then use the matching formula. If the sequence is defined by a rule, compute enough terms to see the pattern before you generalize.

Sequences: complete GMAT guide

Arithmetic sequences

Finding the number of terms

Geometric sequences

Recursive sequences

Patterns and differences

Sum formulas and pairing

Sequences with variables

Alternating and repeating patterns

Sequences and averages

A repeatable approach

Common traps

Quick reference

Practice Problem Solving: Sequences

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