# Formula for the number of terms being compounded

• Square1
In summary: 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.
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 :)

Square1 said:
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...

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?

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

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

Square1 said:
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 ...?

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

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.

## 1. What is the formula for calculating the number of terms being compounded?

The formula for calculating the number of terms being compounded is N = P(1 + r)^n, where N is the final amount, P is the principal amount, r is the annual interest rate, and n is the number of compounding periods.

## 2. How do you use the formula to calculate the number of terms being compounded?

To use the formula, simply plug in the values for N, P, r, and n and solve for the unknown variable. Make sure to use the same units for all values (e.g. if the interest rate is given as a percentage, convert it to a decimal before plugging it into the formula).

## 3. Can the formula be used for any type of compounding?

Yes, the formula can be used for both simple and compound interest compounding. However, for simple interest, the value of r will be constant for all compounding periods, while for compound interest, the value of r may change depending on the compounding frequency.

## 4. What does the value of n represent in the formula?

The value of n represents the number of compounding periods. This can be the number of years for annual compounding, the number of quarters for quarterly compounding, or the number of months for monthly compounding.

## 5. How can the formula for the number of terms being compounded be helpful?

The formula can be helpful in calculating the final amount of an investment or loan, as well as understanding the impact of compounding frequency and interest rate on the final amount. It can also be used to compare different investment or loan options to determine which one will yield the highest return or lowest interest cost.

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