Question

1- find the common ratio and seventh term of the following sequence: 2÷9 ,2÷3,2,6,18....
2-the first term of a geometric series is 1 and the fourth term is 27.find the sum of the first terms.

Answer 1

1.

$r=\frac{a_n}{a_{n-1}}=\frac{a_4}{a_3}=\frac{6}{2}=3.\\ a_1=\frac{2}{9},\\ a_7=a_1r^{7-1}=\frac{2}{9}*3^6=162.$

2.

$a_1=1,\;\;a_4=a_1r^3=27,\\ r=3.\\ S_n=a_a\frac{1-r^n}{1-r}=\frac{1-3^n}{1-3}=\frac{3^n-1}{2}.\\ S_4=\frac{3^4-1}{2}=40.$