# GMAT Math: Progressions

**Introduction**

Sequence and series is one of the most interesting topics for GMAT test takers. A progression consists of a series of numbers in a given order. A constant relation exists between every pair of consecutive terms of a sequence. In general, there are three types of progressions.

1) Arithmetic progression

2) Geometric progression

3) Harmonic progression

In the GMAT, arithmetic and geometric progressions will play a key role. So let us discuss them in detail.

**Arithmetic Progression**

Arithmetic progression is a sequence of numbers in which the difference between any pair of consecutive numbers is same.

• Example: 4,7,10,13,16,19,22,25,28...

The difference between any two consecutive terms in the above sequence is 3, making this an arithmetic progression.

**Property of arithmetic sequence**

**Nth term of arithmetic progression**

is the nth term.

is the first term.

is the nth term.

is the common difference between any two consecutive terms.

**Example:**

Find the nth term of the following sequence: 3, 8, 13, 18, 23, 28, 33, 38...

Find the 100^{th} term of this sequence.

The 100^{th} term of the sequence is 503.

**Sum of n terms of arithmetic progression**

is the last term and is the first term

Or

**Geometric Progression**

A geometric progression is a sequence in which every pair of consecutive terms will produce a common ratio. In other words, every term is obtained by multiplying the previous term by a constant number.

Here, *r* is the common ratio.

are the first, second, third, and nth terms respectively.

**Geometric progression sequence**

This is how a geometric progression looks with respect to the first term.

**Nth term of the geometric progression**

**Example:**

Given the geometric sequence 3, 6, 12, 24, 48...

Find the 10^{th} term of the sequence.

Here the common ratio is 6/3 = 2.

The 10^{th} term will be:

**Sum of n terms of a geometric progression**

The sum of n terms of a geometric progression is given by

**Sum to infinite geometric progression**

Geometric progression has the special feature of allowing you to find the sum to infinity. If the number of terms is infinite, then the sum of the terms looks like this:

**PROBLEM:**

Find the number of terms in the geometric progression 7, 14, 28, 56...7168

The last term is 7868 . Finding which this term is will tell you the total number of terms in the sequence.

The nth term is given by

The common ratio is 14/7 = 2.

So r = 2

Divide by 7 on both sides:

Multiply by 2 on both sides:

2048 is

So

As bases are equal, we can equate the exponents. Hence n = 11.

So the number of terms in the sequence is 11. The following sample problem shows that these can be both easy and interesting to solve.

**GMAT PROBLEM:**

Given a set X that contains the following elements: {y, 0, 3, 6, 9, 12, 15}. What is the value of y if the set contains only whole numbers?

1) -3

2) 18

3) 3

4) Either -3 or 18

5) Doesn't exist

**Solution:**

Let us remove 'y' and look at the sequence. The difference between each term is 3. Hence we can say that this is an arithmetic progression. So as y is the term just before 0 and we may say that 'y' will be 3 less than that, so we can choose y as -3. But it is given that the set contains only whole numbers. So -3 cannot be the possible option. Nowhere in the question is it mentioned that the terms are in order . We can also say that the term can also be after 15. Hence we can say y is 3 greater than 15. That is 18.

So the answer is 18. Option (**B**)

### Related Articles:

- GMAT Math:Picking Numbers
- GMAT Math: Calculation Shortcuts
- GMAT Math: Arithmetic Mean
- GMAT Statistics: Standard Deviation
- GMAT Statistics: Weighted Average and Evenly Spaced Sets

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