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?
Lv 7
? asked in Science & MathematicsMathematics · 6 days ago

The solution an algebraic equation.?

Find the time t. A=P(1+r/n)^(nt), where

A=the amount after nt

P=the initial deposit

r=the annual rate of interest

n=the number of intervals in a year & n->infinity

when A=2P? Help please.

4 Answers

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  • ?
    Lv 7
    6 days ago
    Favourite answer

    It would be useful to memorize the following:

    lim (1 + 1/x)^x = e

    x→∞ 

    let x = n/r

    (1+r/n)^(nt) = (1 + 1/x)^(xrt)

    = ((1 + 1/x)^(x))^(rt)

    lim ((1 + 1/x)^(x))^(rt) = e^(rt)

    x→∞

    Thus

    2P = Pe^(rt)

    t = ln(2)/r

    Because ln(2) is approximately 70% (and less-frequent compounding increases t),  this leads to the "Rule of 72": for a given rate of return r%, your investment will double in approximately 70/r or 72/r years.

  • ?
    Lv 7
    6 days ago

    A=P(1+r/n)^(nt)

    =>

    ln(A)=ln(P)+ntln(1+r/n)

    =>

    ln(A/P)=ln(1+r/n)/[1/(nt)]

    =>

    ln(A/P)->[1/(1+r/n)](-r/n^2)/[(-1/n^2)/t]

    as n->infinity

    =>

    ln(A/P)->rt

    =>

    A=Pe^(rt)

    When A=2P

    2P=Pe^(rt)

    =>

    t=ln(2)/r

    [Note that dependence of memorizing

    formula is not a goodway because there

    are a lot , a lot of formula in math! It  is

    a good habit to derive formula as much

    as possible when needed. Of course,

    some basic formulas are needed to

    remember as the basis]

  • 6 days ago

    I'm not sure what you are asking.

    If you are saying to solve for t from:

    A = P(1 + r/n)^(nt)

    and A = 2P, then we can do the substitution first, then divide both sides by P:

    2P = P(1 + r/n)^(nt)

    2 = (1 + r/n)^(nt)

    Now get the log of both sides:

    ln(2) = ln[(1 + r/n)^(nt)]

    we can pull the exponent out of the log:

    ln(2) = nt ln(1 + r/n)

    And divide both sides by [n ln(1 + r/n)]:

    ln(2) / [n ln(1 + r/n)] = t

  • 6 days ago

     A=P(1+r/n)^(nt)

    2P = P(1 + r/n)^(nt)

    (1 + r/n)^(nt) = 2P/P

    (1 + r/n)^(nt) = 2

    ntln(1 + r/n) = ln(2)

    ...........ln(2)

    t = ------------------

    ........nln(1 + r/n)

    ........... ln(2)

    t = ----------------------- Answer//

    .......... nln[(n + r)/n]

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