Algebra equation with variable as exponent

In summary, the conversation discusses an algebraic method to solve for "r" in the equation V/T = r(1-r^n)/(1-r). It is suggested to use Newton's method and the Excel RATE function. The correct discount factor for a payment in perpetuity is r-1, and for a finite series of payments it is Σ(1+r)-n with n going from first to last payment. Alternatively, one can switch to log returns and continuous compounding to avoid using exponents.
  • #1
adamaero
109
1
This equation takes a present value (PV) to find mortgage payments, PMT:
1665697695309.png


Alternatively, switching V for PV and T for PMT:
V/T = r(1-r^n)/(1-r)

What is an algebraic method to solve for "r"?
Can it not be solved for? I realize I can just find out "r" by trial by error in Excel using the PMT function.

Although, I would like to find a way to just solve for "r" outright. Thanks.
 
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  • #2
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  • #3
I
anuttarasammyak said:
As text says computtion by Newton's method https://en.wikipedia.org/wiki/Newton's_method seems applicable.

It is just a random picture. Newton's method can be punched into an excel spreadsheet?

How?
 
  • #4
adamaero said:
Alternatively, switching V for PV and T for PMT:
V/T = r(1-r^n)/(1-r)
Let me write it in order to use ordinary x-y graph picture
[tex]y=f(x)=\frac{x(1-x^n)}{1-x}-a[/tex]
where n is not a variable but a given number and
[tex]a=\frac{PV}{PMT}[/tex]
is also a given number. The problem is to find x where y=0.

The prescription of Newton method :

Preparation
1 . Get formula of derivative f'(x)
2 . Get formula of tangential line at (x,y=f(x)) with 1.
3 . Get formula of x where the tangential line cross with x-axis with 2, say g(x).

Then
Let x=##x_0## which you assume to be approximate solution
Get value of ##x_1=g(x_0)##
Get value of ##x_2=g(x_1)##
Get value of ##x_3=g(x_2)##
----

Repeat it until you find ##x_n## seem to remain unchanged in PC calcualtion that means ##f(x_n)## is enough close to zero in your calculation environment.
 
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  • #5
I suspect that you are looking for the Excel RATE function, which is described as follows:
Description

Returns the interest rate per period of an annuity. RATE is calculated by iteration and can have zero or more solutions. If the successive results of RATE do not converge to within 0.0000001 after 20 iterations, RATE returns the #NUM! error value.

Syntax

RATE(nper, pmt, pv, [fv], [type], [guess])
 
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Likes anuttarasammyak, PeroK and adamaero
  • #6
pasmith said:
I suspect that you are looking for the Excel RATE function, which is described as follows:

Thank you!
 
  • #7
THe formula in the OP does not look right. for n=1 and r=.05 it gives a discount factor of (.05)(.95)/.95 = .05 the correct factor is (1+r)-1=.95

The discount factor of a payment in perpetuity is simply

r-1

A finite series of payments is simply the sum of each payment's PV, so the PV of a stream of $1 payments is

Σ(1+r)-n with n going from first to last payment
 
  • #8
you can also get rid of the exponents by switching to log returns and continuous compounding

so
(1+r)-n = exp[-rn] where r = log(1+r)
 

FAQ: Algebra equation with variable as exponent

What is an algebra equation with a variable as an exponent?

An algebra equation with a variable as an exponent is an equation in which the unknown quantity is written as a power or exponent. This means that the quantity is being multiplied by itself a certain number of times, where the number of times is represented by the exponent.

How do I solve an algebra equation with a variable as an exponent?

To solve an algebra equation with a variable as an exponent, you can use logarithms or rewrite the equation in a different form. If using logarithms, you would take the logarithm of both sides of the equation. If rewriting the equation, you would try to isolate the variable by using inverse operations.

What are the rules for working with exponents in algebra equations?

The rules for working with exponents in algebra equations include the product rule, quotient rule, power rule, and zero and negative exponent rules. The product rule states that when multiplying two terms with the same base, you can add the exponents. The quotient rule states that when dividing two terms with the same base, you can subtract the exponents. The power rule states that when raising a power to another power, you can multiply the exponents. The zero and negative exponent rules state that any number raised to the power of 0 is equal to 1, and any number raised to a negative exponent can be rewritten as the reciprocal of the number raised to the positive exponent.

What are some real-life applications of algebra equations with variables as exponents?

Algebra equations with variables as exponents are used in various fields such as finance, physics, and chemistry. For example, compound interest calculations use the formula A = P(1+r)^t, where A is the final amount, P is the principal amount, r is the interest rate, and t is the number of compounding periods. In physics, the equation for radioactive decay is N = N0e^(-λt), where N is the final amount, N0 is the initial amount, λ is the decay constant, and t is the time. In chemistry, the ideal gas law equation is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

What are some common mistakes to avoid when solving algebra equations with variables as exponents?

Some common mistakes to avoid when solving algebra equations with variables as exponents include forgetting to apply the rules of exponents, not using parentheses when needed, and not checking the final answer. It is important to remember the rules of exponents and apply them correctly. Parentheses are also important to use when there are multiple terms being raised to an exponent or when using logarithms. Lastly, it is crucial to check the final answer to make sure it makes sense and satisfies the original equation.

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