# Solving Number Series Questions

Number series questions appear on most High School exams and Placement Test. An example is:

Consider the following series: 26, 21, , 11, 6. What is the missing number?

a. 27

b. 23

c. 16

d. 29

Looking carefully at the sequence, we can see right away that each number is 5 less than the previous number, so the missing number is 16.

We can re-write this sequence in mathematical notation as, a_{1}, a_{2}, a_{3},… a_{n}, where n is an integer and a_{n} is called its n^{th} term. And we can write the sequence in the form of a formula, where an integer is substituted in the place of the variable in the formula and the terms are obtained.

For example, let us consider the sequence 5,10,15,20,…

- Here, a
_{n}= 5n. The formula a_{n}= 5n. - The n
^{th}term of a sequence can be found by plugging n in the explicit formula for the sequence. So for example if we wanted to find the 100^{th}number in this sequence, we would substitute n=100 in the formula and get 500.

**Types of Number Sequence problems **

**1. Simple addition or subtraction **– each number in the sequence is obtained by adding a number to the previous number.

For example, 2, 5, 8, 11, 14

Each number in the sequence is obtained by adding 3 to the previous number, which we could write as, a_{n+1} = a_{n} + 3.

**2. Simple multiplication – **each number in the sequence is obtained by multiplying the previous number by a whole number or fraction.

For example, 3, 6, 18, 54

Or,

20, 10, 5, 2.5

Each number in the first sequence is obtained by multiplying the previous number by 3, which we could write as, a_{n+1} = a_{n} X 3.

In the second example, each number in the series is the previous number divided by 2, or multiplied by ½, or a_{n+1} = a_{n} X 1/2.

**3. Prime Numbers – **each number in the sequence is a prime number.

For example,

23, , 31, 37

Answer: 29

**4. Operations on the previous two numbers**

For example,

8, 14, 22, 36, 58

Here the sequence is created by adding the previous 2 numbers.

**5. Exponents**

The number sequence is created by each number squared or cubed.

For example,

3, 9, 81, 6561, where each number is squared.

**6. Combining Sequences
**

2, 7, 13, 20, 28, 37

Here the sequence starts with 2, and each element is added to another sequence starting with 5. So, 2 + 5 = 7, 7 + 6 = 13, 13 + 7 = 20 and so on.

A variation is a sequence with a repeating element. For example,

1, 2, 3, 5, 7, 9, 12, 15

Here the sequence is, for each n, +1, +1, +1, +2, +2, +2, +3, +3,

**7. Fractions
**

For example,

**16/4, 4/2, 2/2, ½, ?**

Fractions are often meant to confuse. If fractions don`t have an obvious relationship, reduce them to lowest terms or to whole numbers. Reducing these to whole numbers, gives,

4, 2, 1, ½

Right away we can see the numbers are half the previous number, so the next in the series is ¼.

In this example, the answer is a fraction, however, you may have to reduce fractions to see the relation, and then convert back to get the answer in the correct form.

Strategy for Answering Number Series Questions

Strategy for Answering Number Series Questions

Answering number series questions is a skill of recognizing patterns, and the best way to improve is to familiarize yourself with the different types, and to practice.

Here is a quick method that will help you answer number series.

For example:

2, 5, 6, 7, 8, ?

**Step 1** – glance at the series quickly and see if you can spot the pattern right away.

**Step 2** – Start analyzing.

Take the different between the first 2 numbers and the different between the second 2 numbers.

2, (+3) 5, (+1) 6, (+1) 7, (+1) 8,

No clear patter with a simple analysis. There is no addition, subtraction, multiplication, division, fractional or exponent relationship.

The relation must be a higher order or a second series.

Next look at the relation between the 1^{st} number and the 2^{nd} and the 1^{st} and the 3^{rd}. We see that,

1^{st} + 3 = 5, 1^{st} + 4 = 6. That’s it! The number 2 is added to the sequence, 3, 4, 5, 6, so the next number will be 2 + 7 = 9.

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