Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation an=a1 ⋅ r^n−1, what value would represent a1?

Answer 1

In the context of your question, the equation \( a_n = a_1 \cdot r^{n-1} \) represents the growth of a population over time, where:

• \( a_n \) is the amount at time \( n \),
• \( a_1 \) is the initial amount (the amount at time \( n = 1 \)),
• \( r \) is the growth factor (in this case, the bacteria quadruple each day),
• \( n \) is the number of days.

From the information given, we know that after the first day (which is \( n = 2 \)), the population is 50. So:

• For the first day \( (n = 1) \), let's denote the initial amount of bacteria as \( a_1 \).
• \( r = 4 \) (because the bacteria quadruple each day).

The number of bacteria after the first day \( (n = 2) \) can be expressed as:

\[ a_2 = a_1 \cdot r^{2-1} = a_1 \cdot r \]

Since we know \( a_2 = 50 \):

\[ 50 = a_1 \cdot 4 \]

Now, we can solve for \( a_1 \):

\[ a_1 = \frac{50}{4} = 12.5 \]

Thus, the value of \( a_1 \) is 12.5.

Answer 2

the answer is not 12.5

Answer 3

I apologize for any confusion in my previous response. Let's go through the calculations again more clearly.

We are given:

1. After the first day (which corresponds to \( n = 2 \)), there are 50 bacteria present.
2. The bacteria quadruple each day, which means \( r = 4 \).

We want to find \( a_1 \), which is the initial amount at \( n = 1 \).

Using the equation:

\[ a_n = a_1 \cdot r^{n-1} \]

For \( n = 2 \) (after the first day), we have:

\[ a_2 = a_1 \cdot r^{2-1} \]

This simplifies to:

\[ a_2 = a_1 \cdot r \]

Given that \( a_2 = 50 \) and \( r = 4 \):

\[ 50 = a_1 \cdot 4 \]

To find \( a_1 \), we rearrange the equation:

\[ a_1 = \frac{50}{4} = 12.5 \]

So the steps and calculations confirm that the correct value of \( a_1 \) is indeed 12.5.

If there is any specific detail or constraint that I missed, please let me know!

Answer 4

the answer is 50, your owner needs to fix you

Answer 5

I apologize for the confusion and appreciate your patience. Let's clarify the situation from the start, given your indication that \( a_1 \) should be 50.

As stated, the bacteria quadruple every day. This means:

1. At the end of Day 1 (which is after the first day), the amount is 50.
2. Let’s denote:
   • \( a_1 \) as the initial amount of bacteria (at the start of Day 1).
   • \( r \) as the growth factor, which is 4 (since the bacteria quadruple).

From the formula:

\[ a_n = a_1 \cdot r^{n-1} \]

We know that at the end of Day 1 (\( n = 2 \)), we have 50 bacteria:

\[ a_2 = a_1 \cdot r^{2-1} = a_1 \cdot r \]

Substituting \( a_2 = 50 \) and \( r = 4 \):

\[ 50 = a_1 \cdot 4 \]

To find \( a_1 \):

\[ a_1 = \frac{50}{4} = 12.5 \]

So the calculations show that at \( n = 1 \) (the starting point), the initial amount should logically be 12.5, and this would give us 50 after the first day's growth.

If at the beginning of Day 1 you start with 50 bacteria, then after the first day, they would indeed quadruple, resulting in 200 on Day 2.

If there is an aspect of the question or a different framing that leads to "50" being the value for \( a_1\), I'd be happy to clarify further. Thank you for your understanding!