1. ## Sequences equal?

There are two sequences:
$a_1,a_2,...; \:\:\:\:\:
b_1,b_2,...$
and each $a_i, b_i$ element of the sequences is 2 or 3. We know that:
$\frac 1 a_1 + \frac 1 {a_1a_2} + \frac 1 {a_1a_2a_3} + ... = \frac 1 b_1 + \frac 1 {b_1b_2} + \frac 1 {b_1b_2b_3} + ...$
Prove that the two sequences are equals.

Any help would be appreciated!

2. Originally Posted by doug
There are two sequences:
$a_1,a_2,...; \:\:\:\:\:
b_1,b_2,...$
and each $a_i, b_i$ element of the sequences is 2 or 3. We know that:
$\frac 1 a_1 + \frac 1 {a_1a_2} + \frac 1 {a_1a_2a_3} + ... = \frac 1 b_1 + \frac 1 {b_1b_2} + \frac 1 {b_1b_2b_3} + ...$
Prove that the two sequences are equals.

Any help would be appreciated!

By reductio ad absurdum. First, show that it is enough to suppose $a_1\neq b_1$ (this already is a little nice problem!),

say $a_1=2\,,\,b_1=3$.

Now, show that $\displaystyle{\frac{1}{2}+\frac{1}{2a_2}+\ldots\ge q \frac{3}{4}\,,\,\,and\,\,\frac{1}{3}+\frac{1}{3b_2 }+\ldots\leq \frac{2}{3}}$ and get a contradiction.

Tonio