Question

How long will it take to double your investment of $1000 at a rate of 2.5% compounding continuously?

Answer 1

`color(blue)(~~27.7 \ \ \ "years")`

Explanation:

The formula for compound interest is given as:

`FV=PV(1+r/n)^(nt)`

Where `bb(FV)` is future value, `bb(PV)` is present value, `bbr` is interest rate as a decimal and `bbn` is the compounding period.

Because we need continuous compounding we have to take the limit as `n->oo`

`PVlim_(n->oo)(1+r/n)^(nt)=e^(rt)=PVe^(rt)`

Using this new formula:

`1000e^(0.025t)=2000`

Divide by 1000:

`e^(0.025t)=2`

Taking natural logarithms of both sides:

`0.025tln(e)=ln(2)`

`t=ln(2)/0.025~~27.72588722`

`color(blue)(27.7 \ \ \ "years")`