Question

In which of the following situations do the list 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.

Answer 1

Let the number of bacteria present initially $=x$
Then, the number of bacteria preset after 1 sec $=2\left(x\right)=2x$
Number of bacteria present after 2 seconds $2\left(2x\right)=4x$
Number of bacteria present after 3 seconds $=2\left(4x\right)=8x$
Number of bacteria present after 4 seconds $=2\left(8x\right)=16x$
So, the number of bacteria from starting to end of each second are given by $x,2x,4x,8x,16x,....$
Now $d_1=2x-x=x,,d_2=4x-2x=2x$
As $d_1\ne d_2$, so the list of numbers does not from an A.P.