# Lines in the Plane

This chapter of Concrete Mathematics reads nicely, without issues so far. I pause often, either to think over the text, draw an image, or write down a table/derivation. My imagination finds fiddling with pen and paper helpful.

The authors of the book introduce the term I wasn't aware of: *triangular
numbers*. Each such `Nth`

number equals the sum of **natural numbers** from `1`

to
`N`

. The following table shows the first seven *triangular numbers*.

N | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

sum | 1 | 3 | 6 | 10 | 15 | 21 | 28 |

Also, we are elegantly introduced to **Gauss's** formula for summing up the
first `N`

**natural numbers**.

Reading the chapter, I've got a metaphor born:

Mathematics can be looked at as a large programming language, to the standard library of which new methods/functions (closed-form formulas) are added by the mathematicians of the world.

Speaking of **closed forms**, let's prove — using the method of mathematical
induction — that, for the *recurrence relation*

the *closed-form* expression is

#### Proof

**Base case**. For ,

But by `(1)`

also, hence `(2)`

is true for the base case.

**Inductive step**. For an arbitrary positive number , such that , suppose that

Now we must show that for :

Observe that

as was to be shown.

Hence this establishes `(2)`

.

And we proceed to the next chapter.