Answer:
First term a₁ = 3/2 and common ratio r = 2
Step-by-step explanation:
We need to find the first term and common ratio while we are given third term = 6 and seventh term = 96
Since common ratio is required so, the sequence is geometric sequence
The formula used is: [tex]a_n= a_1r^{n-1}[/tex]
We are given: third term = 6 i,e
[tex]a_3=a_1r(3-1)\\6=a_1r^2----eq(1)[/tex]
Seventh term = 96
[tex]a_7=a_1r(7-1)\\96=a_1r^6----eq(2)[/tex]
Dividing eq(2) and eq(3)
[tex]\frac{96}{6}=\frac{a_1r^6}{a1r2}\\16=\frac{r^6}{r^2}\\16=r^{6-2}\\=> r^4=16\\Taking \ fourth \ root\\ r=2,-2,2i,-2i\\ \ We \ will \ consider \ only \ positive \ value \ of \ r \ i.e \ \textbf{ r= 2}[/tex]
So, Common Ratio r = 2
Finding First term using eq(1)
[tex]a_3=a_1r^2\\6=a_1(2)^2\\6=a_1(4)\\a_1= \frac{6}{4}\\a_1= \frac{3}{2}[/tex]
So, First term a₁ = 3/2 and common ratio r = 2
