RtW

31 December 2021

# 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 1

the closed-form expression is 2

#### 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 3

Now we must show that for : 4

Observe that as was to be shown.

Hence this establishes (2).

And we proceed to the next chapter.

Created by Y.E.T.If you see an error, please report it.