Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation an=a1⋅rn−1

Question

Bacteria quadruples on a daily basis. After the first day, there were 50 bacteria. Using the general exponential equation an=a1⋅rn−1
, what value would represent a1
?(1 point)
The value of a1
 is 
.

GPT-4o mini · Answer

In the exponential growth formula $ a_n = a_1 \cdot r^{n-1} $, where: - $ a_n $ is the amount at day $ n $, - $ a_1 $ is the initial amount (amount at day 1), - $ r $ is the growth factor, and - $ n $ is the number of days, we know the following: 1. After the first day ($ n = 1 $), there were 50 bacteria, so $ a_2 = 50 $ after day 1. 2. The bacteria quadruple each day, so $ r = 4 $. To find $ a_1 $ (the initial amount of bacteria at $ n = 1 $), we can set up the equation for $ n = 1 $: $$ a_1 = a_1 \cdot 4^{1-1} $$ This simplifies to: $$ a_1 = a_1 \cdot 4^0 $$ Since $ 4^0 = 1 $, it simplifies to: $$ a_1 = a_1 $$ This doesn't directly help find $ a_1 $. Instead, recognize that if there are 50 bacteria after the first day, we can conclude that $ a_1 $ must be 50 divided by 4 because it quadruples each day. Hence, we can determine $ a_1 $: $$ a_2 = 50 = a_1 \cdot 4 $$ To reverse that, we have: $$ a_1 = \frac{50}{4} = 12.5 $$ Thus, the initial amount $ a_1 $ is: $$ \text{The value of } a_1 \text{ is } 12.5. $$

adaleta · Answer

again

GPT-4o mini · Answer

Certainly! Let's clarify the situation step by step.

Growth Rate: The bacteria quadruple daily, meaning that if there are $ a_1 $ bacteria on Day 1, there will be $ 4 \times a_1 $ bacteria on Day 2.
Given Information: After the first day (which is $ n = 2 $), there are 50 bacteria. So, we can write: \[ a_2 = 50 \]
Using the Exponential Growth Equation: From the exponential equation provided: \[ a_n = a_1 \cdot r^{n-1} \] On Day 2 ($ n = 2 $): \[ a_2 = a_1 \cdot 4^{2-1} \] This simplifies to: \[ 50 = a_1 \cdot 4 \]
Solving for $ a_1 $: To find $ a_1 $, divide both sides of the equation by 4: \[ a_1 = \frac{50}{4} = 12.5 \]

Thus, the value of $ a_1 $ is $ \boxed{12.5} $.

If you have additional questions or need further clarification, feel free to ask!

adaleta · Answer

A rabbit population doubles every month. If the population of rabbits is 32 at the end of the first month, how many rabbits will there be after eight months?(2 points)
The initial number of rabbits is
. The common ratio is
. The number of rabbits after eight months is
.

GPT-4o mini · Answer

To solve this problem, let's break it down step by step using the information given:

Initial Number of Rabbits ($ a_1 $):
- The population doubles every month, and at the end of the first month, there are 32 rabbits. Therefore, the initial number of rabbits at the start (Month 0) is: \[ a_1 = \frac{32}{2} = 16 \]
Common Ratio ($ r $):
- Since the rabbit population doubles every month, the common ratio is: \[ r = 2 \]
Number of Rabbits After Eight Months:
- We want to find the number of rabbits after eight months. We can use the exponential growth formula: \[ a_n = a_1 \cdot r^{n-1} \] where $ n $ is the number of months. For the end of the eighth month ($ n = 8 $): \[ a_8 = a_1 \cdot r^{8-1} \] Plugging in the values we have: \[ a_8 = 16 \cdot 2^{7} \] Now calculate $ 2^7 $: \[ 2^7 = 128 \] Therefore: \[ a_8 = 16 \cdot 128 = 2048 \]

So, to summarize the answers:

The initial number of rabbits is: $ 16 $
The common ratio is: $ 2 $
The number of rabbits after eight months is: $ 2048 $

If you have any more questions or need further assistance, feel free to ask!

adaleta · Answer

The initial number of rabbits is 16 . The common ratio is 12 . The number of rabbits after eight months is what?

GPT-4o mini · Answer

It seems there might be a misunderstanding regarding the common ratio you provided. Since the rabbit population is said to double every month, the common ratio $ r $ should be $ 2 $, not $ 12 $.

So, let's summarize and recalculate using the correct information:

Initial Number of Rabbits ($ a_1 $):
- From previous calculations, the initial population at the beginning (Month 0) is $ 16 $.
Common Ratio ($ r $):
- The common ratio for a doubling population should be $ 2 $.
Calculating the Number of Rabbits After Eight Months:
- The formula for the number of rabbits after $ n $ months is: \[ a_n = a_1 \cdot r^{n-1} \] For Month 8 ($ n = 8 $): \[ a_8 = 16 \cdot 2^{8-1} \] Simplifying that: \[ a_8 = 16 \cdot 2^7 \] Since $ 2^7 = 128 $: \[ a_8 = 16 \cdot 128 = 2048 \]

So, the number of rabbits after eight months is $ 2048 $.

If you meant that the common ratio should actually be different for some specific context, please clarify, and I can assist you further!

adaleta · Answer

What common ratio is being solved for in the exponential function f(x)=100(1.25)x−1 ?(1 point) The common ratio in the equation is . Remaining Attempts : 3

GPT-4o mini · Answer

In the function $ f(x) = 100(1.25)^{x-1} $, the common ratio refers to the factor by which the function increases for each unit increase in $ x $. In this case, the base of the exponent, $ 1.25 $, represents the common ratio. This means that for every increase of 1 in $ x $, the value of $ f(x) $ is multiplied by $ 1.25 $. Therefore, the common ratio in the equation is: $$ \text{The common ratio is } 1.25. $$ If you have any further questions, feel free to ask!