Question Video: Finding the Number of Negative Terms in a Given Arithmetic Sequence Mathematics

Video Transcript

Find the number of negative terms in the arithmetic sequence negative 40, negative 33, negative 26, and so on.

An arithmetic sequence has first term 𝑎 and common difference 𝑑. The first term in our sequence is negative 40. Therefore, 𝑎 equals negative 40. To get from negative 40 to negative 33, we add seven. The same is true to get from negative 33 to negative 26. We can therefore say that 𝑑 is equal to seven or positive seven. The general term of any arithmetic sequence is given by 𝑎 plus 𝑛 minus one multiplied by 𝑑.

In this question, we are looking for the number of negative terms. These are the terms that will be less than zero. Our general term equation must be less than zero. Substituting in our values of 𝑎 and 𝑑 gives us negative 40 plus 𝑛 minus one multiplied by seven is less than zero. We can distribute the parentheses or expand the brackets by multiplying seven by 𝑛 and seven by negative one. This gives us negative 40 plus seven 𝑛 minus seven is less than zero.

Adding 40 and seven to both sides of the inequality gives us seven 𝑛 is less than 47. Dividing both sides by seven gives us 𝑛 is less than 47 over seven. As 47 divided by seven is six remainder five, this can be written as a mixed number as six and five-sevenths. 𝑛 must be a whole number or integer value. Therefore, there are six terms that are negative.

We could’ve found these by continuing to add seven to each of the terms in the sequence. Negative 26 add seven is negative 19. Repeating this twice more gives us negative 12 and negative five. The next number in the sequence would be two, which is positive. This once again proves that we have six negative terms in the sequence negative 40, negative 33, negative 26, and so on.