# What is the formula for the #n#th term of the sequence #1, 3, 6, 10, 15,...# ?

##### 2 Answers

#### Explanation:

These are triangular numbers:

#0color(white)(0000)0color(white)(000000)0color(white)(00000000)0color(white)(0000000000)0#

#color(white)(0000)0color(white)(0)0color(white)(0000)0color(white)(0)0color(white)(000000)0color(white)(0)0color(white)(00000000)0color(white)(0)0#

#color(white)(0000000000)0color(white)(0)0color(white)(0)0color(white)(0000)0color(white)(0)0color(white)(0)0color(white)(000000)0color(white)(0)0color(white)(0)0#

#color(white)(000000000000000000)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0000)0color(white)(0)0color(white)(0)0color(white)(0)0#

#color(white)(0000000000000000000000000000)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0#

Geometrically, you can see that such a triangle is one half of a parallelogram with base

#color(white)(0000)0color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)#

#color(white)(000)0color(white)(0)0color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)#

#color(white)(00)0color(white)(0)0color(white)(0)0color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)#

#color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)color(blue)(0)color(white)(0)color(blue)(0)#

#0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)color(blue)(0)#

Such a parallelogram has a total count of

In other words:

#a_n = sum_(k=1)^n k = 1/2n(n+1)#

#### Explanation:

Starting with 1 you can begin to see a pattern develop....

Since every term is merely the sum of the previous term plus the next counting number we have.

To test this we use the first formula for

by the hypothesis

They are equal...

So the hypothesis is proven

This is an example of "Weak" Induction