# Quod Erat Demonstrandum

## 2014/02/17

### 無聊改卷後

Filed under: NSS — johnmayhk @ 10:40 下午

This is a standard question in a quiz:

Solve

$3^x=324-3^{x-1}$

One of my students gave the following so-called solution:

$3^x=3^5+3^4-3^{x-1}$

$x=5+4-x+1$

$x=5$

However, the method is invalid.

## 4 則迴響 »

1. First step is valid though by “wild guessing"; but second step cannot be deduced from first step, right?

迴響 由 Current Chan — 2014/02/17 @ 11:04 下午 | 回應

• $3^x=3^a+3^b-3^{x-1}$ does not imply $x=a+b-x+1$ in general

迴響 由 johnmayhk — 2014/02/17 @ 11:31 下午 | 回應

2. yea, like what I mean

迴響 由 Current Chan — 2014/02/18 @ 12:04 上午 | 回應

3. Stop! And let me raise a defense for the suspect.

$3^x =3^a +3^{a-1} -3^{x-1}$ DOES imply $x= a$ and HENCE $x=a+(a-1)-(x-1)$
So for what reason do you claim that that was invalid?
The “evidence" of the accusation is that he had deduced from the wrong statement.
But firstly, the wrong statement was made explicitly by you. Secondly, the so-called evidence is a guess. This benefit of doubt should invariably go to the defendant.

Okay, I stop now and please forgive my making a joke above.
What is worth discussing here is not the reason for which you determine that method was invalid,
but the reason for which the student made that mistake,
and, practically, how we can let him know that
(1) $n^a =n^b \Rightarrow a=b$ is correct, but
(2) $n^a = n^b+n^c \Rightarrow a=b+c$, as well as its friends, is generally wrong.
and how we can prevent him from making the same mistake again.
I think an un-example on (1) like this questions in the quiz and an example on (2) would be necessary.

迴響 由 CHIN — 2014/02/23 @ 5:18 下午 | 回應