# A strain of bacteria doubles every 14 h. If there are 100 bacteria cells to start with in a colony, how many will there be in 7 days?

A strain of bacteria doubles every $14$ h. If there are $100$ bacteria cells to start with in a colony, how many will there be in $7$ days?

This is a sequence question.

My answer: We start with $100$ cells, so $a = 100$. We double, so $\text{common ratio} = 2$. The formula will be

$$100\cdot 2^{n-1}=100\cdot 2^{14-1}=100\cdot 2^{13}$$

My answer sheet says I'm incorrect, as the answer is $100\cdot 2^{14}$, not $100\cdot 2^{13}$, why is this so? How do I know if I must use the exponent $n-1$, or $n$? I'm confused. Some examples end up being to the $n$-th power, and some end up being to the $(n-1)$-th power.

I'm having a basically identical problem to this person: How To Know in a Application Sequence/Series Problem Which Variable is $a_{0}$ or $a_{1}$?

Thank you!

Both your answer sheet and you are wrong. $7$ days is $168$ hours, which is $12$ periods of $14$ hours, so the number doubles $12$ times. The number at the end is then $100 \cdot 2^{12}$
To tell whether it is $n$ or $n-1$, you need to think about how the problem is stated. The ratio is applied in the gaps, so think about how many gaps there are. If the problem says "on the first day ...", you start counting with $1$. If you count up to $n$, there are $n-1$ gaps, so the power will be $n-1$. In this case, we are told that at the starting moment there are $100 \cdot 2^0$ bacteria and then we go through $12$ intervals.
• That is correct. If you call the first term $a_0$, then the general term is $a_n=a_0+nd$ – Ross Millikan Jun 17 '15 at 2:05