Finding the Sum of a Finite Number of Terms for t = 64/(165+3n)

[izzleshizzum]

Homework Statement

I want a general equation for the finite sum of n₀ + n₁ + n₂... starting at n = 0 for the equation t = 64/(165+3n) so i have a sum of numbers: 64/165 + 64/168 + 64/171...

i don't want you to think i am lazy and don't show work but this isn't for school. i want to figure out how much time having passed on a very long recording of a metronome corresponds to the beats per minute being played.

1 bar = 4 beats
start at 55% of 300 beats per minute
after 16 bars have passed it starts over but 3 beats per minute are added to the speed.
so it goes from 165 beats per minute to 168 to 171...

i could just add them all up and make a little reference table but i am determined now to understand how to solve this problem!

Homework Equations

t = 64/(165+3n)

The Attempt at a Solution

my attempts are all useless.

 

[icystrike]

Can you express it as:
$$\frac{64}{3}\sum_{n=0}^{i}\frac{1}{55+n}$$
Such that :
$$i \geq n$$
$$i \in\mathbb{Z}^{+}$$

 

[Gib Z]

Welcome to PF!

So when you say "finite" sum, are you summing a finite amount of terms, or summing an infinite number of terms but want a finite result? Because the sum is actually divergent ie it does not sum to any finite number.

If you are however summing a finite number of terms, even a large amount, it can be done exactly by some Mathematics Program such as Maple or Mathematica, or approximated quite well with some simple analysis.