Question

The interest is supposed compound unless the contrary is state.
If $a,b,c$ years' purchase must be paid for an annuity to continue $n,2n,3n$ years respectively; show that
$a^2-ab+b^2=ac$.

Answer 1

Let $a,b,c$ be the present value of the annuity for $n,2n,3n$ years respectively.

$a=\frac{1-r^{-n}}{r-1}$ where $r$ be the interest rate .

$b=\frac{1-r^{-2n}}{r-1}$

$c=\frac{1-r^{-3n}}{r-1}$

$\frac{b}{a}=\frac{1-r^{-2n}}{1-r^{-n}}$

$\frac{b}{a}=\frac{(1-r^{-n})(1+r^{-n})}{1-r^{-n}}$

$\frac{b}{a}=1+r^{-n}$

Similarly,

$\frac{c}{a}=\frac{1-r^{-3n}}{1-r^{-n}}$

$\frac{c}{a}=\frac{(1-r^{-n})(1+r^{-n}+r^{-2n})}{1-r^{-n}}$

$\frac{c}{a}=1+r^{-n}+r^{-2n}$

$\frac{c}{a}=1+(\frac{b}{a}-1)+{(\frac{b}{a}-1)}^2$

$\frac{c}{a}=\frac{b}{a}+\frac{b^2}{a^2}+1-\frac{2b}{a}$

$\frac{c}{a}=\frac{b^2}{a^2}+1-\frac{b}{a}$

$a^2-ab+b^2=ac$