# How to Solve Civil Service Exam Number Series Problems 4

This is the fourth part of the solving number series problems. The first part discussed patterns that contains addition and subtraction and the second part discusses patterns that contains multiplication or division. The third part was about alternating patterns.

In this post, we are going to discuss some special number patterns. Although there is a small probability that these types of patterns will appear in the Civil Service Examination (I didn’t see any when I took the exams, both professional and subprofessional), it is better that you know that such patterns exist.

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, …

Triangular numbers are numbers that are formed by arranging dots in triangular patterns. Therefore, the first term is 1, the second term is 1 + 2, the third term is 1 + 2 + 3 and so on.

Square Numbers

1, 4, 9, 16, 25, 36, …

The square numbers is a sequence of perfect squares: $1^2$, $2^2$, $3^2$, $4^2$, $5^2$, $6^2$, and so on.

Cube Numbers

1, 8, 27, 64, 125, …

Well, from square numbers, you surely have guessed what are cube numbers. They are a sequence of cube of integers.

$1^3$, $2^3$, $3^3$, $4^3$, …

Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, …

Technically, a Fibonacci sequence is a sequence that starts with (0, 1), or (1, 1), and each term is the sum of the previous two. For example, in the sequence above, 5 is the sum of 2 and 3, while 21 is the sum of 8 and 13. In the actual examination, they may give Fibonacci-like sequences (technically called Lucas Sequence) where they start with two different numbers. For example, a Lucas sequence that starts with 1 and 3 will generate

1, 3, 4, 7, 11, 18, …

Of course, they can also combine positive and negative numbers to create such sequences. For example, a Lucas sequence that starts with -8 and 3 will generate the sequence

-8, 3, -5, -2, -7, -9, …

Well, this looks like a difficult sequence, but remember that if you can see the pattern, it is easy to look for the next terms.

Images Credit: Math World