Real Sequence Arithmetic and Geometric Progression Problems

Question

At what rate of interest will a single investment triple its value in 5years?

Guys, please help me here, so far I have done this:

Using the formula;

a + (n-1)d

let the first term (a) = 1

Let the nth term = 5

find d?

substituting all the values into the formula I have

1 + (5-1)d 1 + 5d - d = 0 5d - d = 0-1 4d = -1 4d/4 = -1/4

d = -1/4

d = 0.25%

I don't have any idea what I am doing is right, but am I in line?

Answer 1

Simple interest In line with the usual notation for arithmetic progressions, we'll take the convention that time starts at the first term, not zeroth. The investment's value is $a+(n-1)d$ at time $n$, i.e. the investment starts at $n=1$< not at $n=0$. The value grows from $a$ at $n=1$ to $3a=a+5d$ when the investment ends at $n=6$. Rearranging $d=\frac25 a$ i.e. the annual interest rate is $40\%$.

Compound interest Similar logic gives $3a=ar^5$, with $a$ the initial investment value and $r$ the common ratio. Thus $r=3^{1/5}$. Then the interest rate per annum is $100(3^{1/5}-1)\%\approx 24.57\%$.