In which of the following situations, do the lists of numbers involved form an $ A P $ ? Give reasons for your answers.
The number of bacteria in a certain food item after each second, when they double in every second.

Given:

The number of bacteria in a certain food item after each second, when they double in every second.

To do:

We have to check whether  the list of numbers involved in the given situation forms an \( A P \).

Solution:

Let the number of bacteria in a certain food be $x$.

Given that the bacteria double in every second.

This implies,

The number of bacteria after every second is,

$x, 2 x, 2(2 x), 2(2 \times 2 x), \ldots$

The sequence formed is,

$x, 2 x, 4 x, 8 x, \ldots$

In the given sequence,

$a_1=x, a_2=2x, a_3=4x, a_4=8x$

$a_2-a_1=2x-x=x$ 

$a_3-a_2=4x-2x=2x$

$a_4-a_3=8x-4x=4x$

Here,

$a_3-a_2≠a_2-a_1$

Therefore,

The list of numbers involved in the given situation does not form an \( A P \).