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

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In summary, the conversation discusses finding the finite sum of a series of numbers based on a given equation, with the goal of determining the corresponding beats per minute for a recording of a metronome. The equation t = 64/(165+3n) is provided, and the conversation includes a discussion of a possible solution using a reference table or solving the problem mathematically.
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izzleshizzum
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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.
 
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  • #2
Can you express it as:
[tex]\frac{64}{3}\sum_{n=0}^{i}\frac{1}{55+n}[/tex]
Such that :
[tex]i \geq n[/tex]
[tex]i \in\mathbb{Z}^{+}[/tex]
 
Last edited:
  • #3
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.
 

1. What is the formula for calculating the finite sum of T = 64/(165+3n)?

The formula for calculating the finite sum of T = 64/(165+3n) is n/3 * (1/165 + 1/(165+3n)).

2. How do you represent a finite sum in mathematical notation?

A finite sum can be represented using the sigma notation: Σ (k=1 to n) T, where T is the expression or function to be summed and n is the number of terms in the sum.

3. Can the finite sum of T = 64/(165+3n) be simplified?

Yes, the finite sum of T = 64/(165+3n) can be simplified to n/3 * (1/165 + 1/(165+3n)) = n/495 + n/(495+9n) = n/(495+9n).

4. What is the significance of the finite sum in the formula T = 64/(165+3n)?

The finite sum in the formula T = 64/(165+3n) represents the total value or result of the expression when a finite number of terms (n) are added together.

5. How can the finite sum of T = 64/(165+3n) be applied in scientific research or real-world problems?

The finite sum of T = 64/(165+3n) can be used in various applications such as economics, statistics, and physics to calculate the total value or result of a function or expression when a finite number of terms are added together. This can be useful in analyzing data and making predictions in scientific research or solving real-world problems.

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