1. Exponential equation word problems

Hi,

I hope someone can help. I'm trying to solve the following question:

As stated above, the equation is N(t) = 3^(t+1). N(t) represents the number of people told during the hour t, and t = 1. However, let's say that we wanted to develop an equation which fit the same context that represented the total number of people over a given amount of time? Could someone please explain to me what this equation would look like?

I'm trying to develop an understanding for exponential functions, and so I would really appreciate help with this.

Thanks,
Olivia

2. Re: Exponential equation word problems

If I understand correctly you want

\begin{align*} &N=\sum \limits_{k=0}^t ~3^{k+1} = \\ \\ &3 \sum \limits_{k=0}^t ~3^k = \\ \\ &3 \dfrac{3^{t+1}-1}{3-1} = \\ \\ &\dfrac{3}{2} \left(3^{t+1}-1\right),~t \in \mathbb{N} \end{align*}

3. Re: Exponential equation word problems

$\displaystyle \sum_{i=0}^t 3^{i+1} = 3 + 3^2 + 3^3 + ... + 3^{t+1}$

$S_t = 3 + 3^2 + 3^3 + ... + 3^{t+1}$

$3 \cdot S_t = 3^2 + 3^3 + ... + 3^{t+2}$

$3 \cdot S_t - S_t = 3^{t+2} - 3$

$S_t (3-1) = 3^{t+2} - 3$

$S_t = \dfrac{3(3^{t+1}-1)}{2}$

Have you studied geometric series & their sums ... ?

4. Re: Exponential equation word problems

Originally Posted by otownsend
As stated above, the equation is N(t) = 3^(t+1). N(t) represents the number of people told during the hour t, and t = 1. However, let's say that we wanted to develop an equation which fit the same context that represented the total number of people over a given amount of time? Could someone please explain to me what this equation would look like?
Like romsek, I to am confused by what you are are asking. I do not think that it evolves summations. Rather this is more an exercise in careful reading. We are told that $\mathcal{N}(t)$ is the number of people told in the hour $t$. Moreover, we are also told that $\mathcal{N}(t)=3^{t+1}$ and that three people were told during the first hour. Now as counter-intuitive as it may be, that implies that in the first hour $t=0$.
According to the attachment, the exercise asks: "During what hour are 2187 people told?"
We note that $2187=3^7$ see here.
Thus, we have $\mathcal{N}(t)=3^{t+1}=2187$ so $t=~?$.

5. Re: Exponential equation word problems

Unfortunately, I haven't learned geometric series and their sums yet. For now I will think about this question some more on my own and then get back if I'm still unsure.

6. Re: Exponential equation word problems

Originally Posted by otownsend
Unfortunately, I haven't learned geometric series and their sums yet. For now I will think about this question some more on my own and then get back if I'm still unsure.
Did you read my post? The above reinforces my reading. Because you have not learned summations, the question is simply about what geometric sequences are.