Answer:
[tex]\larg\boxed{\large\boxed{57}}[/tex]
Explanation:
1. Write the general general form of the nth term of a geometric series
[tex]a_n=a_1\times r^{n-1};n=1,2,3,...[/tex]
[tex]r=\text{common ratio}[/tex]
[tex]a_1=\text{first term}[/tex]
2. Write the expression for the product of the first five terms equal to 243
[tex]a_1\times a_2\times a_3\times a_4\times a_5=243\\\\a_1\times (a_1\times r)\times (a_1\times r^2)\times (a_1\times r^3)\times (a_1\times r^4)=243[/tex]
[tex]a_{1}^5r^{10}=243[/tex]
[tex](a_{1}r^{2})^5=3^5[/tex]
[tex]a_1r^2=3[/tex]
3. Write the third term
Notice that the third term is [tex]a_1\times r^2[/tex]
Then, the third term is 3.
4. Goe with the aritmetic series: the tenth term is 3
[tex]a_{10}=3[/tex]
5. Write the expression for the sum of the first 19 terms of the arithmetic series
General formula:
[tex]S = (n/2)\times \bigg(2a_1+(n-1)d\bigg)[/tex]
For n = 19
[tex]S = (19/2)\times (2a_1+18d)[/tex]
6. Write the expression for the term #10
[tex]a_{10}=a_1+9d=3\\\\2(a_1+9d)=2(3)\\\\2a_1+18d=6[/tex]
7. Substitute in the expression for S
[tex]S=(19/2)\times 6\\\\S=57\leftarrow answer[/tex]
