Exercise 122 Page 237

We must find the equation and the fifth term $t(5)$ of an arithmetic sequence with the given terms. Recall that an arithmetic sequence is characterized by having an addition or subtraction sequence generator. This is a constant number that is added or subtracted to the term to get the next term. Let's analyze the information we have so far. We can see that the sequence is decreasing, since the $8^{\text{th}}$ term is $1056$ but the $13^{\text{th}}$ term is $116.$ With this information we know that a constant number $c$ is being subtracted. From the $8^{\text{th}}$ term to the $13^{\text{th}}$ term this happens $5$ times, so we can write the equation $t(8)-5c=t(13).$ By solving it we can find the common difference $c.$ Since the common difference from term to term is $188,$ the general form of the equation for this sequence is $t(n)=t(1)-188(n-1).$ We still need to find what $t(1)$ is. We can use one of the values for the terms we know and solve for $t(1).$ Having found the first term, we can write the equation for our sequence as $t(n)=2372-188(n-1).$ Finally, we need to find $t(5).$ Let's evaluate our expression and find this term.