Arithmetic Squence

• Sep 13th 2011, 06:37 PM
NoCrown
Arithmetic Squence
A doctor prescribes 200 mg of a medication on the first day, and the dosage is reduced by half every day, for one week. What is the total amount of medication prescribed?

I know the answer, but the way I got the answer is definitely incorrect. This question was meant to be solved using an arithmetic or geometric equation, and although I know the majority of them, I seem to have forgotten the exact equation for solving a question like this. The correct answer is 397 mg. Can anyone help me?
• Sep 13th 2011, 06:41 PM
TheChaz
Re: Arithmetic Squence
So there are seven days. Of course, if you just add up
200
100
50
25
12.5
6.25
3.125
The way to do this is with the sum

$\displaystyle S_n = \frac{n}{2}(a_1 + a_n)$

When n is the number of terms, and the a's are the first and last terms.

So we have n = 7
a_1 = 200
a_n = a_7 = 3.125

Compute.
• Sep 13th 2011, 06:49 PM
NoCrown
Re: Arithmetic Squence
Thank you very much. Is there an easier way to find the "n"th term?
• Sep 13th 2011, 06:58 PM
TheChaz
Re: Arithmetic Squence
a_n = a_1 + (n - 1)d
Where d is the signed difference between consecutive terms.
• Sep 14th 2011, 06:00 AM
Soroban
Re: Arithmetic Squence
Hello, NoCrown!

Quote:

A doctor prescribes 200 mg of a medication on the first day,
and the dosage is reduced by half every day, for one week.
What is the total amount of medication prescribed?

Um, this is a geometric sequence.

The first term is $\displaystyle a = 200$, the common ratio is $\displaystyle r = \tfrac{1}{2}$

The sum of the first $\displaystyle n$ terms is: .$\displaystyle S_n \:=\:a\,\frac{1-r^n}{1-r}$

We have: .$\displaystyle S_7 \:=\:200\,\frac{1-(\frac{1}{2})^7}{1-\frac{1}{2}} \:=\:200\,\frac{\frac{127}{128}}{\frac{1}{2}} \:=\:\frac{3175}{8} \;=\;396\tfrac{7}{8}\text{ mg.}$
• Sep 14th 2011, 08:12 AM
TheChaz
Re: Arithmetic Squence
Quote:

Originally Posted by Soroban
Hello, NoCrown!

Um, this is a geometric sequence.

The first term is $\displaystyle a = 200$, the common ratio is $\displaystyle r = \tfrac{1}{2}$

The sum of the first $\displaystyle n$ terms is: .$\displaystyle S_n \:=\:a\,\frac{1-r^n}{1-r}$

We have: .$\displaystyle S_7 \:=\:200\,\frac{1-(\frac{1}{2})^7}{1-\frac{1}{2}} \:=\:200\,\frac{\frac{127}{128}}{\frac{1}{2}} \:=\:\frac{3175}{8} \;=\;396\tfrac{7}{8}\text{ mg.}$

I have disgraced my family!
(commits hari-kari)