The formula in your equation needs grouping symbols, such as parentheses,
to make it correct:
A = P(1 + r/n)^(nt)
Otherwise, a quantity would have been raised to n, and THEN multiplied by t.
With the version I typed, this will make sure the quantity will be raised to the
product nt.
= . . . . . = . . . . . = . . . . . = . . . . . = . . . . . = . . . . . = . . . . . = . . . . . =
As far as my knowledge goes, it is not possible to isolate n, but you can
approximate isolating n.
A = P(1 + r/n)^(nt)
A/P = (1 + r/n)^(nt)
Take specifically natural logs of each side:
ln(A/P) = ln(1 + r/n)^(nt)
ln(A/P) = (nt)ln(1 + r/n)
Note: ln(1 + r/n) is approximately equal to
r/n  (1/2)(r/n)^2 + (1/3)(r/n)^3  ...
Continuing,
ln(A/P) ~ (nt)[r/n  (1/2)(r/n)^2 + (1/3)(r/n)^3]
Use the corresponding equation to this approximation.
Distribute the n from inside the parentheses into the brackets:
ln(A/P) = (t)[r  r^2/(2n) + r^3/(3n^2)]
Distribute the t and clear the brackets:
ln(A/P) = tr  tr^2/(2n) + tr^3/(3n^2)
ln(A/P)  tr = tr^2/(2n) + tr^3/(3n^2)
Multiply both sides by the L.C.D. of 6n^2:
[ln(A/P)  tr]6n^2 = 3tnr^2 + 2tr^3
Get all the terms to one side:
[ln(A/P)  tr](6)n^2 + 3tnr^2  2tr^3 = 0
n = {3tr^2 +/ sqrt[9(t^2)(r^4)  4(ln(A/P)  tr)(6)(2tr^3)]}/{2[ln(A/P)  tr](6)}
EDIT: More simplifying in this quadratic formula could be done/shown,
with certain constants being multiplied together and certain constants
and/or variables having their order changed.
There should be one legitimate value here,
and the rounded value for n should be an
integer such as (but not limited to)
1, 2, 4, 12, 52, or 365 to correspond,
respectively to, annually, semiannually,
quarterly, monthly, weekly, or daily.
