Question Video: Solving Word Problems Involving Arithmetic Sequences Mathematics

Video Transcript

A doctor prescribed 15 pills for his patient to be taken in the first week. Given that the patient should decrease the dosage by three pills every week, find the week in which he will stop taking the medicine completely.

We’re told in the question that the doctor prescribed 15 pills for the patient to take in the first week. The patient should then decrease the dosage, that means the number of pills they take, by three pills every week. That means in the second week, they should take three less than 15, 15 minus three which is equal to 12. In the third week, they’ll take three less again, 12 minus three which is equal to nine, so nine pills in the third week.

We want to find the week in which the patient will stop taking the medicine completely. So we need to continue like this until we get to zero pills. In the fourth week, the patient will take nine minus three, which is equal to six pills; in the fifth week, six minus three, which is three pills; and finally, in the sixth week, three minus three, which is zero. So in the sixth week, the patient will stop taking the medicine completely.

Another way to approach this problem would be to find a formula for the general term in the sequence. As the terms decrease by the same amount each time, this means the difference between successive terms is constant. And so this is an example of an arithmetic sequence. The general term of an arithmetic sequence is given by 𝑎 sub 𝑛 equals 𝑎 plus 𝑛 minus one 𝑑, where 𝑎 sub 𝑛 represents the 𝑛th term. 𝑎 or sometimes 𝑎 one represents the first term. 𝑛 represents the term number. And 𝑑 represents the common difference between the terms.

The first term in this sequence is 15, and the common difference is negative three. So the formula for the general term is 𝑎 sub 𝑛 equals 15 minus three multiplied by 𝑛 minus one. To find the week in which the patient will stop taking the medicine completely, we can set the general term equal to zero, because that represents the number of pills, and then solve the resulting equation to find 𝑛, the term number.

First, we can add three multiplied by 𝑛 minus one to each side, giving three multiplied by 𝑛 minus one is equal to 15. We can then divide both sides of the equation by three, giving one multiplied by 𝑛 minus one or simply 𝑛 minus one is equal to five. We then add one to each side of the equation giving 𝑛 equals six. So using a different method, we’ve found that the number of pills will be zero in the sixth week, which confirms that the patient will stop taking the medicine completely in the sixth week.