There are four main categories of Sequences that appear GMAT Quantitative Section: Arithmetic, Geometric, Repeating and Sums. Think of sequences as a simple pattern, and detecting this pattern is probably the most difficult part.

### Arithmetic Sequences

An Arithmetic Sequence is when subsequent terms in a sequence increase (or decrease) by a constant amount. Here’s the standard formula:

- = + (n – 1)d, where is the value at term n, and d is the constant change.

Note that – = d

If given the following sequence, we can derive both _{ }and d, to solve for any term.

8, 11, 14, 17, 20, 23….. ( = 8 and d = 3)

So, if asked what term number 86 is, we can just plug in to the formula:

= + (n – 1)d

= 8 + (86 – 1)3 = 8 + 85*3 = 8 + 255 = 263

### Geometric Sequences

The same principle applies to Geometric Sequences, in which each subsequent term is *multiplied *by a certain constant. Compound interest in an example of a geometric sequence. Here’s the standard formula.

** = **** *** , where is the value at term n, and r is multiplicative rate of increase.

Note that / = r** **

Similarly, when given the rate of increase and the value of any term, we can find any other term. For example:* *

*John originally put $8 in his piggy bank in 2001. If his parents double the money in the piggy bank once a year (without John adding anymore himself), how much money will he have in 2009?*

r = 2

**= ** *

= 8

= 8*

n = 8

= $1,024

### Repeating Patterns

Many times, repeating patterns will yield a remainder question.* *

*The first term of a sequence is -2 and the second term is 2. Each subsequent odd term is found by adding 2 to the previous term, and each subsequent even terms is found by multiplying the previous term by -1. What is the sum of the first 669 terms?*

* *

Clearly, we are not looking to enter all 669 terms and see what the last one is. But, we can do a few and check out the pattern.

n = 1, = -2

n = 2, = 2

n = 3, = 4

n = 4, = -4

n = 5, = -2

n = 6, = 2

Since we can see that the pattern will repeat every 4 terms, we can solve for the remainder after dividing 669/4. Since 4 goes into 668 evenly, we know that the value will be equivalent to that of the first term, which = -2.

### Sums

** **

With sequence questions involving sums, identify the pattern. This is the most important first step toward finding the solution. Check out the following example.* *

*A set of consecutive integers begins at -19. After which term will the sum of all the terms equal 41?*

* *

Again, we don’t want to write all these out. BUT, we can start, and try to see if we can detect the pattern. In this particular example, we will be adding integers in the negative space until 0, after which we will be adding the positive pair of the same integers all the way up to +19. The sum after +19 equals zero, equals 20 after +20 and equals 41 after +21.

That means 19 negative terms + 1 (zero) + 21 positive terms = 41 total terms.