Suppose that i sub $n = 5%$ for all $n ≥ 1$. How long will it take an investment to triple in value?

Question

I tried to use $3x=x(e^{.05t})$

then i got $\ln3+\ln x=\ln x+.05t$

so $t=\ln(3)/.05 = 21.97$

but i know that the answer is $22.5171$

Please use MathJax (https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) for future post. — Arnaldo, May 17 '17 at 13:50

score 2 · Answer 1 · edited May 17 '17 at 14:33

2

The discrepancy in the value is due to the use of continuous compounding than the discrete compounding.

So $3X = X(1+.05)^t$

Taking logarithm on both sides, you get $\ln(3) = t\ln(1.05)$

$t = \ln(3)/\ln(1.05) = 22.5171$

There you go!!

I still wonder what n is? and x_n is ?

Goodluck

edited May 17 '17 at 14:33

Arnaldo

answered May 17 '17 at 14:12

Satish Ramanathan

score 1 · Answer 2 · edited May 12 '20 at 17:12

You will use the formula

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

where: $A$ = accumulated amount $P$ = investment $t$ = time $r$ = interest rate $n$ = number of times compounded per year

$$P = P$$ $$A = 3P$$
$$r = 0.05$$ $$n = 1$$ $$t = 1$$

Substituting these values into the formula,

$$3P = P(1 + 0.05)t$$

Divide by $P$ on both sides of the equation.

$$3 = 1.05t$$

Log both sides of the equation

$$\log(3) = \log(1.05)t$$

Bring the exponent t down as the coefficient of the log(1.05)

$$\log(3) = t \log(1.05)$$

Divide by $log(1.05)$ into both sides of the equation

$$t = \log(3) / \log(1.05)$$ $$t = 22.52$$

It will take $22.52$ years to triple the investment at an interest rate of $5$%.

2 Answers2