The first and last terms of an arithmetic sequence are negative 55 and 209, respectively. There are 21 terms between the first and last terms. Find the list of these intermediate terms.
We begin by recalling that an arithmetic sequence has a common difference between consecutive terms. The general term or 𝑛th term in an arithmetic sequence can be found using the formula 𝑎 sub 𝑛 is equal to 𝑎 sub one plus 𝑛 minus one multiplied by 𝑑, where 𝑎 sub one is the first term in the sequence and 𝑑 is the common difference. In this question, we are told that the first term in the sequence is negative 55. The last term in the sequence is 209. As there are 21 terms between them, there are 21 plus two or 23 terms altogether in the sequence. This means that the last term 209 is the 23rd term.
We can substitute these values into the general formula to calculate the common difference 𝑑. The 23rd term is equal to negative 55 plus 23 minus one multiplied by 𝑑. And as this term equals 209, we have 209 is equal to negative 55 plus 22𝑑. Adding 55 to both sides of this equation, we have 22𝑑 is equal to 264. We can then divide through by 22, giving us an answer for the common difference 𝑑 equal to 12.
As the first term was negative 55, we need to add 12 to this to find the next term. This is equal to negative 43. Repeating this process, the next two terms are negative 31 and negative 19. We know that the last term in the sequence is 209. As the common difference is 12, the term before this will be 12 less than 209. This is equal to 197. The entire sequence is negative 55, negative 43, negative 31, negative 19, and so on, up to 197 and 209. This is not the final answer, however, as we were just asked to list the intermediate terms. This is the list of terms between the first and last term.
The answer to this question is the sequence of numbers negative 43, negative 31, negative 19, and so on, all the way up to 197.