Question

Let a1 = 1, b1 = 2 and c1 = 3. Consider the convergent sequences
\left\\{a_n\right\\}^{\infin}_{n=1},\left\\{b_n\right\\}^{\infin}_{n=1} \text{ and }\left\\{c_n\right\\}^{\infin}_{n=1}
defined as follows :
$a_{n+1}=\frac{a_n+b_n}{2},b_{n+1}=\frac{b_n+c_n}{2} \text{ and } c_{n+1}=\frac{c_n+a_n}{2} \text{ for } n \ge1.$
Then,
$\sum\limits_{n=1}^{\infin}b_nc_n(a_{n+1}-a_n)+\sum\limits_{n=1}^{\infin}(b_{n+1}c_{n+1}-b_nc_n)a_{n+1}$
equals _____________ (rounded off to two decimal places)

Answer 1

The correct answer is 1.95 to 2.05. (approx)