Step 1
Given;
[tex]The\text{ series; 5,14,23}[/tex]Required; To find the sum of the first 31 terms
Step 2
The series is arithemtic, hence the common difference(d) will be
[tex]\begin{gathered} T_2-T_1=14-5=9 \\ d=9 \end{gathered}[/tex]Step 3
Find the sum of the first 31 terms
[tex]Sum\text{ of terms of an A.p is given as; S}_n=\frac{n}{2}(2a+(n-1)d)[/tex][tex]\begin{gathered} n=31 \\ d=9 \\ a_=5 \\ S_{31}=\frac{31}{2}(2(5)+(31-1)9) \\ \end{gathered}[/tex][tex]\begin{gathered} S_{31}=\frac{31}{2}(10+30(9)) \\ S_{31}=\frac{31}{2}(280)=4340 \end{gathered}[/tex]Answer;
[tex]S_{31}=4340[/tex]