# How to solve sequence questions in GMAT Quant

Applying rules to questions involving sequences is often the fastest way to solve such problems.

Consider the following question from GMAT Tutor's bank of practice questions:

*In a sequence of 19 numbers, each term except for the first one is greater than the previous term by 9. If the last term in the sequence is 200, what is the first term in the sequence?*

## The trick to solving this sequence question quickly

In a sequence the nth term can be expressed as (the first term) + (the difference between the terms) multiplied by (n-1). In mathematical terms this rule can be expressed as:

**A _{n} = A_{1} + (n - 1)d**

Here, d represents the difference between the terms.

If you spend some time thinking about the logical nature of this rule, it will help you to remember it under pressure on test day.

- Imagine you have the sequence 1,3,5.
- The difference between the terms is 2.
- The third term is the sum of the first term + (n-1) x 2
- This is 1 + 2 x 2
- 1 + 4 = 5

If you forget any rule on test day, try to derive the rule in the way mentioned above. Set up a simple situation and work out the rule from there.

## Using the rule to solve this sequence problem

Let’s apply this rule to the question we are considering.

- A
_{n}= A_{1}+ (n - 1)d - Thus, to find the first term, we
**rearrange**: A_{1}= A_{n}- (n - 1)d - A
_{1}= 200 – 18 x 9 - A
_{1}= 200 – 162 - A
_{1}= 38

Note that this rule allows you to determine the value of any term of the sequence, as long as you have one term in the sequence and the difference between the terms.

Now that you know how to identify numbers in a sequence, let's try another type of question involving sequences: finding the median.

**Consider the following question from GMAT Tutor:**

*If M is the set of all consecutive multiples of 9 between 100 and 500, what is the median of M?*

## How to solve this sequence question

- Establish the first and last numbers in the set.
- You should know in your head that 12 x 9 = 108. If you don’t, this is an opportunity to realise that knowing your times tables in your head is one of the most useful time-saving methods on the GMAT. 108 is therefore the first number in our set, as it is a multiple of 9 and greater than 100.
- Divide 500 by 9 and you get a little more than 55.
- Multiply 55 x 9. You get 495. The last number in our set is therefore 495.

## An important rule for sequences

Now here comes the the trick. The important rule for you to remember is that the median of a sequence of numbers is the same as the average. Secondly, the average in a sequence of numbers can be obtained by adding the first and last terms of the sequence and dividing by 2.

**Therefore:**

- 108 + 495 = 603
- 603 / 2 = 301.5
- Therefore, 301.5 is the median of Set M.

## Plugging in numbers to solve sequence questions

Don’t forget the strategy of plugging in numbers when solving sequence questions. This strategy can be used in most areas of GMAT Quant, and it applies to sequences, as well.

**Consider the following question from GMAT Tutor:**

*The total number of plums that grow during each year on a certain plum tree is equal to the number of plums that grew during the previous year, less the age of the tree in years (rounded down to the nearest integer). During its 3rd year, the plum tree grew 50 plums. If this trend continues, how many plums will it grow during its 6th year?*

**Plug in the numbers:**

- In its fourth year the tree grew 50 - 3 = 47 plums.
- (Notice that this is not 50 - 4, because the age of the tree is still 3 years, not yet 4 years.)
- Continue this method
- 5th year: 47 - 4 = 43
- 6th year 43 – 5 = 38
- The answer is therefore 38.

Remember these simple rules and strategies about sequences and you’ll have more time for the next question on the Quant section!