Question

Triangular numbers can be represented with equilateral triangles formed by dots. The first five triangular numbers are 1, 3, 6, 10, and 15. Is there a direct variation between a triangular number and its position in the sequence? Explain your reasoning.

Answer 1

Answer:

Step-by-step explanation:

Triangular Number Sequence

This is the Triangular Number Sequence:

1, 3, 6, 10, 15, 21, 28, 36, 45, ...

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

triangular numbers

By adding another row of dots and counting all the dots we can

find the next number of the sequence.

The first triangle has just one dot.

The second triangle has another row with 2 extra dots, making 1 + 2 = 3

The third triangle has another row with 3 extra dots, making 1 + 2 + 3 = 6

The fourth has 1 + 2 + 3 + 4 = 10

etc!

How may dots in the 60th triangle?

A Rule

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

First, rearrange the dots like this:

triangular numbers 1 to 5

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

triangular numbers when doubled become n by n+1 rectangles

Now it is easy to work out how many dots: just multiply n by n+1

Dots in rectangle = n(n+1)

But remember we doubled the number of dots, so

Dots in triangle = n(n+1)/2

We can use xn to mean "dots in triangle n", so we get the rule:

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?

log stack

Example: You are stacking logs.

There is enough ground for you to lay 22 logs side-by-side.

How many logs can you fit in the stack?

x22 = 22(22+1)/2 = 253

You may get tired, and the stack may be dangerously high, but you can fit 253 logs in it!

Answer 2

Answer:

No, the triangular numbers are not a direct variation. There is not a constant of variation between a number and its position in the sequence. The ratios of the numbers to their positions are not equal. Also, the points (1, 1), (2, 3), (3, 6), and so on, do not lie on a line.

Step-by-step explanation: