Question

Spot the first step where a mistake occurs in the solution of the problem given below. (Options might be in random order, Step 1 is the first step of the solution and Step 4 is the last step of the solution) Step 1: S=\sum^{4}_{x=1} log(1/x) Step 2: S=log(1/1)+log(1/2)+log(1/3)+log(1/4) Step 3: S=log(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}) Step 4: S=log(\frac{25}{12})

Answer 1

To find:

$S=\sum^{4}_{x=1} log(1/x)$

Expanding the above statement:

Step 2:Taking sum of the series from x=1 to x=4

Putting x=1, the term is : $log(\dfrac{1}{1})$

Putting x=2, the term is : $log(\dfrac{1}{2})$

Putting x=3, the term is : $log(\dfrac{1}{3})$

Putting x=4, the term is : $log(\dfrac{1}{4})$

Now,

Step 2:

$S=log(\dfrac{1}{1})+log(\dfrac{1}{2})+log(\dfrac{1}{3})+log(\dfrac{1}{4})$

Formula: $loga+logb = log(ab)$

This formula is applicable for more than 2 terms as well i.e.

$loga+logb+logc+logd = log(abcd)$

Applying the formula:

Step 3:

$S=log[(\dfrac{1}{1})\times (\dfrac{1}{2})\times (\dfrac{1}{3})\times (\dfrac{1}{4})]$

So, this step has a mistake.

As per question statement, at this step:

$S=log(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4})$

Step 4: Actual answer will be:

$S=log[(\frac{1}{1 \times 2 \times 3 \times 4})]\\\Rightarrow S = log(\frac{1}{24})$

instead of $S=log(\frac{25}{12})$