Sequences and Series | Sequences and Series for MAT PDF Download

Sequence

• 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.

Series

• A series can be highly generalized as the sum of all the terms in a sequence.
• It is a finite or infinite sequence of numbers that are added together according to a specific pattern or rule.  

Sequence vs Series

• A sequence is an ordered list of numbers or terms that follow a specific pattern or rule.
• Each term in a sequence is assigned a position or index that represents its order in the sequence.
• A series is the sum of the terms in a sequence.
• It is the result obtained by adding up the individual terms of a sequence.

Types of Sequences 

(i) Geometric Sequences: 

(ii) Arithmetic Sequences: 

Types of Series

There are two types of Series

(i) Geometric Series: 

• A geometric series is a series in which each term is obtained by multiplying the previous term by a fixed number called the common ratio (r).
• The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.

(ii) Arithmetic Series: 

• An arithmetic series is a series in which each term is obtained by adding a fixed number called the common difference (d) to the previous term.
• The general form of an arithmetic series is a + (a + d) + (a + 2d) + (a + 3d) + ..., where a is the first
  term and d is a common difference.

Geometric Sequence vs Arithmetic Sequence

1. Arithmetic sequence: 

• In 'set A', the common difference is the fixed amount of one. In 'set B' the common difference is the fixed amount of two, and in 'set C' the common difference is the fixed amount of five. 
• As you most likely noticed already, the common difference is found by finding the difference between two consecutive terms within the sequence. 
• For example, in 'set C', to find the common difference compute (8 - 3 = 5).

2. Geometric sequence: 

Set D= {2,4,8,16,32}

Set E= {3,9,27,81}

Set F= {5,10,20,40,80}

You might notice that the difference between consecutive numbers in the above three sets is not a fixed amount.

For instance, in 'set F', the first two terms (5 and 10) have a smaller difference than the last two terms (40 and 80). Therefore, the above sets are geometric sequences.

The difference between two consecutive numbers is therefore the common ratio. To find the common ratio you simply take the ratio of one consecutive number to the one before it. In 'set F' this would be (10/5=2). Therefore, n 'set F' the common ratio is two. In 'set E' the common ratio is (27/9=3). In 'set D' the common ratio is two (32/16=2).

Summary: The difference between the two types of sequences is that in arithmetic sequences the consecutive numbers in a set differ by a fixed amount known as the common difference whereas in a geometric sequence, the consecutive numbers in a set differ by a fixed number known as the common ratio.

Geometric series vs Arithmetic series

1. Geometric series:

• A geometric series is a series in which each term is obtained by multiplying the previous term by a fixed number called the common ratio (r).
• The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where "a" is the first term and "r" is the common ratio.
• Geometric series can be finite (having a specific number of terms) or infinite (extending indefinitely).
• The sum of a finite geometric series can be calculated using the formula:
  Sum = a * (1 - r^n) / (1 - r), where "n" is the number of terms.
• For an infinite geometric series with |r| < 1 (the absolute value of the common ratio is less than 1), the sum can be found using the formula:
  Sum = a / (1 - r).

2. Arithmetic series:

• An arithmetic series is a series in which each term is obtained by adding a fixed number called the common difference (d) to the previous term.
• The general form of an arithmetic series is a + (a + d) + (a + 2d) + (a + 3d) + ..., where "a" is the first term and "d" is the common difference.
• Arithmetic series can be finite or infinite.
• The sum of a finite arithmetic series can be calculated using the formula:
  Sum = (n/2) * (2a + (n - 1)d), where "n" is the number of terms.

What the GMAT could ask us to do with Sequences and Series and How to do it!

There is no limit to what the GMAT can ask you to find when dealing with series and sequences. Here are some examples of things you may be asked to find/do with them.

(1) The sum of numbers in a series (which can be asked in many tricky ways such as the sum of all the numbers, the sum of just the even numbers, the sum of just the odd numbers, the sum of only the numbers which are multiples of 7, the sum of the first 10 numbers, and many more tricky ways!)

(2) The nth term in a sequence.

(3) How many integers are there in a sequence

Anyway, now that you get the point... let's give you the formulas that will allow you to answer any question regarding series and sequences. I will then show you how to use the formulas to answer some questions that might not be intuitive to nonmath geniuses.

The Formula for Geometric Sequence (when there is a Common Ratio)

A geometric sequence is a list of numbers, where the next term of the sequence is found by multiplying the term by a constant, called the common ratio.
The general form of the geometric sequence formula is
where r is the common ratio, a1 is the first term, and n is the placement of the term in the sequence.

The Formula for Arithmetic Sequence (when there is a Common Difference)

The formula for finding the nth term (the term at position n) in an arithmetic sequence is:

nth term = a + (n - 1) * d

In this formula:

"a" represents the first term of the sequence.

"d" represents the common difference between consecutive terms.

"n" represents the position of the term you want to find.

Example: 
Let's consider an arithmetic sequence with a first term (a) of 3 and a common difference (d) of 2. If we want to find the value of the 5th term (n = 5), we can use the formula:
5th term = 3 + (5 - 1) * 2
= 3 + 4 * 2
= 3 + 8
= 11
So, the 5th term of the sequence with a first term of 3 and a common difference of 2 is 11.

The Formula for Geometric Series (when there is a Common Ratio)

The formula for the sum of a finite geometric series (when there is a common ratio) is as follows:

Sum = a * (1 - r^n) / (1 - r)

In this formula:

• "Sum" represents the sum of the geometric series.
• "a" represents the first term of the series.
• "r" represents the common ratio between consecutive terms.
• "n" represents the number of terms in the series.

Example: 
Let's consider a geometric series with a first term (a) of 2, a common ratio (r) of 3, and a number of terms (n) of 4.
Using the formula, we can find the sum of this series:
Sum = 2 * (1 - 3^4) / (1 - 3) = 2 * (1 - 81) / (1 - 3)
= 2 * (-80) / (-2)
= 80

So, the sum of the geometric series with a first term of 2, a common ratio of 3, and 4 terms is 80.

The formula for Arithmetic series (when there is a Common Difference) 

• The formula for the sum of an arithmetic series (when there is a common difference) is as follows:
  S_n = n/2[2a + (n-1)d]
  S_n = sum of the series
  a1 = the first term
  an = the nth term
  n = the number of terms
  d = the common difference between consecutive terms
• Example: 
  Let's consider an arithmetic series with a first term (a) of 2, a common difference (d) of 3, and a number of terms (n) of 4. Using the formula, we can find the sum of this series:

Sum = (4/2) * (2 + (4-1)3) = 2 * (2 + 33) = 2 * (2 + 9) = 2 * 11 = 22

So, the sum of the arithmetic series with a first term of 2, a common difference of 3, and 4 terms is 22. 

How to find the number of Integers in a set

• *A mistake is that people will forget to add the 1. The number of terms between 3 and 10 is not 7, it is 8. A common mistake is that people will calculate (10-3=7)... but this is wrong. Remember, as Manhattan GMAT says, "Add one before you are done".
• *Notice how I used the word "term" and not a number. This is important because sometimes you don't always just put the first and last number you are given. For example, If you are asked to find the number of even integers between 1 and 30, you don't use the "first number" in the set. The first number is "1", which is odd, and we are only speaking about even numbers. Therefore, the first term is "2", not "1", even though the set or question might have stated, "from 1-30". The same goes for the last term.

There is another step needed to answer this question though.

Find the number of odd integers (or even) in a set

• *If the question is to find the number of odd integers between 2 and 30, then your first term is 3, and your last term is 29. They must be odd to fit in the set you are asked to analyze.
• *If the question is to find the number of even integers between 3 and 29, then your first term is 4, and your last term is 28.

Find the number of integers that are a multiple of a certain number in a set

GMAT questions can get tricky, but luckily not too tricky. For example... What if you are asked to "find the number of multiples of 7 between 2 and 120"?

All you have to do is instead of dividing our old formula by 2, you divide it by the increment.
Also, notice how my first and last terms are the first term which is a multiple of 7 and the last term which is a multiple of seven within the set!

Find the sum of odd numbers in a series

This seems to be a popular topic on GMAT forums. It's quite simple. You already know everything you need to after reading this post. It is a two-step problem. Here are the two steps:

(1) Find the number of odd terms. This is your "n" value now.

(2) Plug in the "n" value into the formula for an arithmetic series.

There are some shortcuts and concepts that you should know about this topic:

(1) The mean and the medium of any arithmetic sequence are equal to the average of the first and last terms.
(2) The sum of an arithmetic sequence is equal to the mean (average) times the number of terms.
(3) The product of n consecutive integers is always divisible by n! So, 4x5x6 (4*5*6=120) is divisible by 3!
(4) If you have an odd number of terms in a consecutive set, the sum of those numbers is divisible by the number of terms.
(5) number four (above) does not hold true for consecutive sets with an even amount of terms.