Math Is Fun Forum

Hixy

Member

Registered: 2011-09-24

Posts: 15

Analysis - analyzing the input of a function

First the warm-up problem:

My reasoning:

Is this the correct way to look at it? Can the function be found somehow?

Now to the problem:

Any hints as to how I can progress with this? I've tried isolating f(n) and find the starting point of the function, but that is as far as my thought process has brought me at the moment.

Hixy

Member

Registered: 2011-09-24

Posts: 15

Re: Analysis - analyzing the input of a function

Just realized I might have something. Hang on.

Edit:
Okay I got it.

Rewrite the original expression as

is an arithmetic series (=the sum of an arithmetic progression). The nth term can be expressed as

, with d being the difference between each term.
By double counting, the entire sum

can be expressed as

In (2) the sum is expressed in terms of

, and in (3) in terms of

. Counting the same set twice is indeed a useful technique!
Adding both sides and dividing by 2 gives

Since

, it follows

Finding the functions for

and inserting in (1) reduces the entire thing to

Arithmetic series wiki: http://en.wikipedia.org/wiki/Arithmetic_progression
Double counting wiki: http://en.wikipedia.org/wiki/Double_counting_(proof_technique)

Last edited by Hixy (2011-10-15 09:40:51)