1. ## Geometric Sequences

Hello. I was hoping someone could help me with this problem I have? Anything you guys can help me with would be greatly appreciated.

A certain radioactive isotope has a half-life of 11 days. If one is to make a table showing the half-life decay of a sample of this isotope from 32 grams to 1 gram; list the time (in days, starting with t = 0) in the first column and the mass remaining (in grams) in the second column, which type of sequence is used in the first column and which type of sequence is used in the second column?

2. ## Re: Geometric Sequences

I recommend the following.

1. Learn the definitions of arithmetic and geometric progressions.
2. Make the required table where the first column contains not every day, but every 11th day.

After you do this, you should know the answer.

3. ## Re: Geometric Sequences

Thanks for your help. I think I got it, but will you check my answer for me?

In the first column you are trying to calculate the time starting at 0. Since the number of days is increasing, you want to add n number of days to the equation. But, you started at zero days so you need to subtract one from the increments. So you get a final equation of tn=n-1. Since the zero makes virtually no diff

In the second column you would set the equation up with Mass (m) equals the starting mass (32) divided by 2 to the n-1 of a degree. You want to use the mass that the isotope starts with and work your way down by half (since it is half-life). Since time starts at zero days, you want to be sure to subtract one from the nth power as to count for the time. So the equation looks like mn=32/2^n-1.

4. ## Re: Geometric Sequences

If you make a table with every day, then the mass is divided not by 2, but by $\sqrt[11]{2}$. On the other hand, if you divide the mass by 2, then the table must contain every 11th day.

In any case, the question asks to identify the type of the sequence, not to come up with the formula for the nth element, doesn't it?

5. ## Re: Geometric Sequences

Oh. Well I'm an idiot.

The first sequence would be geometric and the second would be arithmetic right?

6. ## Re: Geometric Sequences

Originally Posted by Wald
The first sequence would be geometric and the second would be arithmetic right?
No. Check again the definitions of arithmetic and geometric sequences.

7. ## Re: Geometric Sequences

Ok, so since the first sequence has a difference of just 1, it is arithmetic?

And since the second sequence decreases by half every single time, it is geometric?

8. ## Re: Geometric Sequences

Yes, except, as I said, it's either the difference 1 and the ratio $\sqrt[11]{1/2}$ or the difference 11 and the ratio 1/2.

9. ## Re: Geometric Sequences

So if I say that the first sequence is arithmetic because its difference is 11, then I have to say that the second sequence is decreasing by a ratio of 1/2. Its just one slightly changes how the other sequence is perceived?

10. ## Re: Geometric Sequences

I suggested writing every 11th day for simplicity, as a preliminary step to answering the question. Since the half-life is 11 days, the mass is multiplied by 1/2, so it's easy to figure out that the second column contains a geometric sequence. One does not have to find the ratio for each day, which is $\sqrt[11]{1/2}$. Since the original question is so simple, I thought that figuring out this ratio may be a problem and a distraction.

However, the question talks about only one table, where the first column is an arithmetic sequence with difference 1 and the second column is a geometric sequence with ratio $\sqrt[11]{1/2}$.

11. ## Re: Geometric Sequences

I see what you mean now.

Thank so much for your help. I really do appreciate your helping me.