Formula for the number of terms being compounded

[Square1]

Ran out of room for my title - not sure how to say it concisely, or if there is a better place to put this question...lets say this is "financial engineering"...

I am looking for a formula what will give me the number of terms that will be compounded in a given length of time.

Say:
Principle = 1000
i = 10%

If we are including the principle as a term, for the end of the first period T1 we have the original principle accruing interest. At the end of the second period T2, now we have two terms accruing interest (the principle and the previous interest earned). At the end of the third period T3 we have four terms accruing interest. At the end of T4, I think we got nine terms accruing interest. T1 = 1, T2 = 2, T3 = 4, T4 = 9. Similarily, if we exclude the principle term, the pattern is like this: T1 = 0, T2 = 1, T3 = 3, T4 = 8.

So, how would you go about getting a formula for the number of terms? Does anyone know what it is?

Wow it's been perhaps 3 years or so since I've last posted here. Oh the memories coming back :) Love you all :)

 

[billy_joule]

If we are including the principle as a term, for the end of the first period T1 we have the original principle accruing interest. At the end of the second period T2, now we have two terms accruing interest (the principle and the previous interest earned). At the end of the third period T3 we have four terms accruing interest. At the end of T4, I think we got nine terms accruing interest. T1 = 1, T2 = 2, T3 = 4, T4 = 9. Similarily, if we exclude the principle term, the pattern is like this: T1 = 0, T2 = 1, T3 = 3, T4 = 8.

So, how would you go about getting a formula for the number of terms? Does anyone know what it is?

If I'm understanding your explanation right, your series are wrong. At the end of each period every term accrues interest which results in a doubling of the number of interest earning terms ie

1,2,4,8,16,32, 64 ...

(rather than: T1 = 1, T2 = 2, T3 = 4, T4 = 9)

Where an expression to find the value of the nth term can be had quite simply...

 

[Square1]

OK, yes, I made a mistake and it automatically makes things much easier...Yes I was asking for is the amount of terms getting compounded in any year (previous interest and the principle). Since I should have had T4 = 8 instead of 9 including principle, your set of numbers are what I am looking for.

The next thing I wanted to think about is, a formula for the total amount of payments made over a given amount of time. It appears to be T1 = 1, T2 = 3, T3 = 7, T4 = 15, T5 = 31...I googled and found (2^n) -1. Are there other expressions? Something that is recursive?

I ask this because I'm just looking to quantify the amount of payments coming out of an investment that compounds, vs one that doesn't...(2^n) -1 vs. n...

Now having said that, this is all related to the compound interest formula P(1+i)^n. Is (2^n) -1, where n is for # of payments, derivable from P(1+i)^n, or are this completely separate matters?

 

[SteamKing]

It's "principal", not "principle", as in the "principal amount" (the amount you first start with).

This article discusses some of the math behind compound interest:

http://en.wikipedia.org/wiki/Compound_interest

OK, yes, I made a mistake and it automatically makes things much easier...Yes I was asking for is the amount of terms getting compounded in any year (previous interest and the principle). Since I should have had T4 = 8 instead of 9 including principle, your set of numbers are what I am looking for.

The next thing I wanted to think about is, a formula for the total amount of payments made over a given amount of time. It appears to be T1 = 1, T2 = 3, T3 = 7, T4 = 15, T5 = 31...I googled and found (2^n) -1. Are there other expressions? Something that is recursive?

I ask this because I'm just looking to quantify the amount of payments coming out of an investment that compounds, vs one that doesn't...(2^n) -1 vs. n...

Now having said that, this is all related to the compound interest formula P(1+i)^n. Is (2^n) -1, where n is for # of payments, derivable from P(1+i)^n, or are this completely separate matters?

I'm not sure, but it sounds like you are asking about how to figure how much money you will get out of some kind of annuity, versus drawing periodically from a fixed sum.

Annuities are discussed here:

http://en.wikipedia.org/wiki/Annuity_(finance_theory)

Perhaps if you explained in more detail what you are looking for ...?

 

[Square1]

It's "principal", not "principle", as in the "principal amount" (the amount you first start with).

This article discusses some of the math behind compound interest:

http://en.wikipedia.org/wiki/Compound_interest
I'm not sure, but it sounds like you are asking about how to figure how much money you will get out of some kind of annuity, versus drawing periodically from a fixed sum.

Annuities are discussed here:

http://en.wikipedia.org/wiki/Annuity_(finance_theory)

Perhaps if you explained in more detail what you are looking for ...?

Thanks for the spelling tip :)

My initial questions about getting a formula for the number of payments is solved.

At this point I am wondering what strategy/strategies would be used to derive the formula for the number of total interest payments after given number of periods...just a general overview. It appears to me that applying recursive thinking may be one way. (2^n) - 1 seems like it would be derived another way (which I'm not sure of how). Furthermore, how closely would such formulas be related to the formula for compound interest p(1+i)^n? I.e. is there any math that can be done directly on p(1+i)^n to find these formulas, or as I've mentioned, do you start for the most part with a blank card?

Hope this is clearer. I'm not a math student so my literacy in it is not the best.