Computing The Arithmetic-Harmonic Mean

We notice that

$a_n\times b_n = a_0 \times b_0 = 100 \quad \forall n \in Z^{+} \ldots equation\{1\}$

We plot the sequences $\{a_n\}$ and $\{b_n\}$ on the number line.

Thus for given points

$a_n$ and $b_n$ ( $b_n<a_n, n\geq 1$ ) , $a_{n+1}$ is the midpoint of $a_n$ and $b_n$ while $b_{n+1}$ is less than $a_{n+1}$ (since H.M.<A.M.) and is closer to $b_n$ than to $a_n$ .

Continuing further, $a_{n+2}$ is the midpoint of $a_{n+1}$ and $b_{n+1}$ and similarly $b_{n+2}$ is closer to $b_{n+1}$ than $a_{n+1}$ .

We observe that $a_i$ and $b_i$ come close to each other as $i$ increases and both sequences converge to a single point when $i$ becomes large. Hence, we concude:

$\displaystyle \lim_{n\rightarrow \infty} a_n = \displaystyle \lim_{n\rightarrow \infty} b_n = k$

From the equation $\{1\}$ , we get:

$k^2 = 100$

$\displaystyle \Rightarrow \boxed{k=10}$