Given: An arithmetic sequaence has the following parameters
[tex]\begin{gathered} a_9=17 \\ d=-2.1 \end{gathered}[/tex]To Determine: The sum of the first 31st term.
Please note that the sum of the first 31st term is represented as
[tex]S_{31}=\text{ sum of the first 31st term}[/tex]The formula for the finding the n-term of an arithmetic sequence (AP) is
[tex]\begin{gathered} a_n=a+(n-1)d \\ \text{Where} \\ a_n=n-\text{term} \\ a=\text{first term} \\ d=\text{common difference} \end{gathered}[/tex]Since, we are given the 9th term as 17, we can calculate the first term a, as shown below:
[tex]\begin{gathered} a_9=17 \\ \text{Substituting into the formula} \\ a_9=a+(9-1)d \\ a_9=a+8d \\ \text{Therefore:} \\ a+8d=17 \\ d=-2.1 \\ a+8(-2.1)=17 \\ a-16.8=17 \\ a=17+16.8 \\ a=33.8 \end{gathered}[/tex]Calculate the sum of the first 31st term.
The formula for finding the first n-terms of an arithmetic series is given as
[tex]S_n=\frac{n}{2}(2a+(n-1)d)[/tex]We are given the following:
[tex]a=33.8,n=31,d=-2.1[/tex]Substitute the given into the formula:
[tex]\begin{gathered} S_{31}=\frac{31}{2}(2(33.8)+(31-1)-2.1) \\ S_{31}=15.5(67.6)+(30)-2.1) \\ S_{31}=15.5(67.6-63) \end{gathered}[/tex][tex]\begin{gathered} S_{31}=15.5(4.6) \\ S_{31}=71.3 \end{gathered}[/tex]Hence, the sum of the first 31st term of the A.P is 71.3, OPTION D
