Question

the total number of circle in 5rows of triangular number patterns is

Answer 1

This sequence comes from a pattern of dots that form a triangle:



By adding another row of dots and counting all the dots we can 
find the next number of the sequence.

A Rule

We can make a "Rule" so we can calculate any triangular number.

First, rearrange the dots (and give each pattern a number n), like this:



Then double the number of dots, and form them into a rectangle:



The rectangles are n high and n+1 wideand xn is how many dots in the triangle (the value of the Triangular Number n)

And we get (remembering we doubled the dots):

2xn = n(n+1)

xn = n(n+1)/2

Rule: xn = n(n+1)/2

Example: the 5th Triangular Number is

x5 = 5(5+1)/2 = 15

Example: the 60th is

x60 = 60(60+1)/2 = 1830

Wasn't it much easier to use the formula than to add up all those dots?