A sequence is an ordered list of numbers. Each number contained in a sequence is called a term.

A sequence is defined by an equation or rule. The rule can be applied to each term of a sequence to generate the next term in the sequence. The ellipsis (…) at the end of a sequence means the sequence continues using the same rule.

The key to solving sequence problems is to determine the relationship between the terms.

## Example

What is the next term in the sequence?

2, 4, 6, 8, …

### Solution

(a) 2, 4, 6, 8, …

Here you can probably assume the next number of the sequence is 10, but always be careful with snap assumptions on sequence problems.

## Don’t Assume

The GRE likes to trick you with a sequence like:

What is the next term in this sequence? 3, 5, 7, ____, …

You may immediately think “consecutive odd numbers” and assume the next term is 9. But it could be “consecutive primes” and the next term would be 11. On Data Sufficiency questions in particular, be careful not to make assumptions about the rule in a sequence.

#### Writing the Rules

The key to solving sequence problems is to determine the relationship between the terms in the sequence. The rule can be applied to each term of a sequence to generate the next term in that sequence.

There is a common set of variables used in the equation or rule for a sequence.

Subscripts are used to give the position-number of a term.

*a _{n}* represents the value of the

*n*th term.

*d* is the difference between terms.

*r* is the ratio between terms.

So for the sequence 2, 4, 6, 8, 10:

**value**

**term & number**

**pattern**

2

*a*_{1} = 2

*a*_{1} = 2

4

*a*_{2} = 4

*a*_{2} = *a*_{1} + 2 = 4

6

*a*_{3}= 6

*a*_{3} = *a*_{2} + 2 = 6

8

*a*_{4} = 8

*a*_{4} = *a*_{3} + 2 = 8

10

*a*_{5} = 10

*a*_{5} = *a*_{4} + 2 = 10

## Example

Write the first 4 terms of the sequence:

a=_{n}n+ 4

### Solution

Substitute the term number for *n* in the equation that describes the sequence.

The first term *a*_{1} would be *a*_{1} = 1 + 4 = 5.

The second term *a*_{2} would be *a*_{2} = 2 + 4 = 6.

The third term *a*_{3 }would be *a*_{3} = 3 + 4 = 7.

The fourth term *a*_{4 }would be *a*_{4} = 4 + 4 = 8.

## Example

Write the rule for the sequence: 5, 10, 15, 20, 25, …

### Solution

Look for a pattern that uses the position number *n*.

The first term *a*_{1} = 5 = 1 × 5.

The second term *a*_{2 }= 10 = 2 × 5.

The third term *a*_{3} = 15 = 3 × 5.

The fourth term *a*_{4} = 20 = 4 × 5.

So the rule for the sequence is that the *n*th term is *n* × 5.

*a _{n}* = 5

*n*

#### Arithmetic Sequence

In an arithmetic sequence the difference between consecutive terms is constant. You add (or subtract) the same amount to go from one term to the next. From the Example above, *a _{n}* =

*n*+ 4 is an arithmetic sequence.

There is a general form for the formula of an arithmetic sequence. This formula allows you to calculate the value of any term in an arithmetic sequence with just the value of the first term and the difference.

arithmetic sequence *a _{n}* =

*a*+ (

_{1}*n*– 1)

*d*

## Example

Find the 100th term in this sequence: 2, 5, 8, 11, 14, …

### Solution

Sometimes the easiest way to find a term is to just do the calculations.

**term** *a*_{1} *a*_{2} *a*_{3} *a*_{4} *a*_{5} *a*_{6} *a*_{7} * a*_{8} *a*_{9}

**value ** 2 5 8 11 14 17 20 23 26

But for 100 terms, there has to be a better way.

From looking at the first 5 terms, you can see the difference is 3. Use the formula for arithmetic sequence with *a*_{1}= 2 and *d* = 3.

*a*_{100} = 2 + (100 – 1)(3) = 2 + (99)(3) = 299

The 100th term is 299.

## Example

The fifth term in a sequence of numbers is 19. Each term after the first term in the sequence is 3 less than the term immediately preceding it. What is the second term in the sequence?

(A) 10

(B) 13

(C) 28

(D) 30

(E) 31

### Solution

A good technique to visualize this short sequence is to use blanks.

____ ____ ____ ____ ___19___ ____

*a*_{1} *a*_{2 } *a*_{3 } *a*_{4 } *a*_{5 } *a*_{6}

You are told that each term is 3 less than the term immediately preceding it.

Does that mean that the fourth term, the one immediately preceding 19, will be 3 more or 3 less? Be sure to read carefully!

19 is 3 less than the term preceding it. So *a*_{4} – 3 = *a*_{5} = 19, and *a*_{4} = 22.

Continue to fill in the blanks.

___31___ ___28___ ___25___ ___22___ ___19___ ____

*a*_{1 } *a*_{2 } *a*_{3 } *a*_{4 } *a*_{5 } *a*_{6}

So the second term *a*_{2} = 28. The correct answer choice is (C).

#### Arithmetic Series

A series is the sum of the terms in a sequence.

sequence: 5, 10, 15, 20

series: 5 + 10 + 15 + 20 = 50

There is a formula for arithmetic series. The formula allows you to calculate the sum of the arithmetic sequence using the number of terms and the values of the first and last terms.

The sum of an arithmetic sequence is the mean of the first and last terms multiplied by the number of terms.

arithmetic series

*S _{n}* =

*n*(

*a*

_{1}+

*a*/2)

_{n }Notice how similar this is to the formula for adding consecutive integers, such as the sum of all the integers from 1 to 100. The concept here is the same: take the mean of the terms and multiply it by the number of terms.

(Note: This is covered further here: Consecutive Numbers.)

## Example

Find the sum of the first 100 terms in this series: 2 + 5 + 8 + 11 + 14 + …

### Solution

With this many terms, there must be a shortcut.

The first step is to calculate the value of the 100th term. An Example above does this calculation:

Use the formula for arithmetic sequence with *a*_{1} = 2 and* d* = 3.

*a*_{100} = 2 + (100 – 1)(3) = 2 + (99)(3) = 299 The 100th term is 299.

So the first term is 2, the 100th term is 299, and there are 100 terms.

*S*_{100} = 100 (2 + 299/2)=100 (301/2) =100(150.5) = 15,050

#### Geometric Sequence

In a geometric sequence the ratio between consecutive terms is constant. You multiply by the same number to go from one term to the next. From an Example above, *a _{n}* = 5

*n*is a geometric sequence where the ratio is 5.

There is a formula for a geometric sequence that allows you to calculate the value of any term in a geometric sequence with just the value of the first term and the ratio.

geometric sequence *a _{n}* =

*a*

_{1}×

*r*

^{(n-1)}

## Example

Find the 10

^{th}term in the sequence: 3, 6, 12, 24, …

### Solution

First find the ratio between terms.

6/3 = 2

12/6 = 2

24/12 = 2

Use *r* = 2 and *a*_{1} = 3 in the formula.

*a*_{10} = 3 × 2^{(10 − 1)} = 3 × 29 = 3 × 512 = 1,536 The 10th term is 1,536.

## Example

Find the 7th and 8th terms in the sequence: -729, 243, -81, 27, …

### Solution

First find the ratio between terms.

243 / –729 = –1/3

-81 / 243 = –1/3

27 / –81 = –1/3

So *r* = –1/3.

*Method 1*

You could use *a*_{1} = -729 in the formula and look for the 7th and 8th terms. But if you use *a*_{4} = 27 as the first term, the calculations will be easier. You will look for the fourth and fifth terms after 27.

*a*_{4} = 27 × (-1/3)^{(4 − 1)}= 27 × (-1/3)^{3} = 27 × (-1/27) = 27 / -27 = -1

For (-1/3)^{3} remember negative × negative × negative = negative.

*a*_{5} = 27 × (-1/3)^{(5 − 1)} = 27 × (-1/3)^{4} = 27 × (1/81) = 27 / 81 = 1/3

For (-1/3)^{4} remember negative × negative = positive.

*Method 2*

Another technique for this short sequence is to use blanks. This makes it easier to see the alternating positive and negative terms.

___-729___ ___243___ ____-81___ ____27___ ____-9___ ____3____ ____-1___ ___1/3___

*a*_{1 } *a*_{2 } *a*_{3 } *a*_{4 } *a*_{5 } *a*_{6 } *a*_{7 } *a*_{8}

So the 7th and 8th terms are -1 and 1/3.

## Example

Except for the first two terms, every term in the sequence 1, -2, -2, 4, … is the product of the two immediately preceding terms. What is the seventh term of this sequence?

(A) -8

(B) 32

(C) -32

(D) 256

(E) -256

### Solution

*Method 1 *

Rather than a ratio between terms, there is a rule. Though you could write the rule for this short sequence, it is easier to use blanks or a list.

*a*_{1} = 1

*a*_{2} = -2

*a*_{3} = -2

*a*_{4} = 4

*a*_{5} = -2 × 4 = -8

*a*_{6} = 4 × -8 = -32

*a*_{7} = -8 × -32 = 256

The correct answer choice is (D).

*Method 2*

Use the strategy of reading the answers first. You can see that there are two basic pieces of information you need to find: whether the number is positive or negative, and how big it is.

First find the pattern for positive and negative without doing the calculations:

positive, negative, negative, positive, negative, negative, positive.

If you quickly calculate the fifth term, just looking at the change between the fourth term and the fifth terms shows you the value increases quickly. From that, you can infer that the largest value will be correct. The largest positive value is 256, which is (D).