Question

Find the first term and the common ratio
third term =6 and seventh term= 96​

Answer 1

First term a₁ = 3/2 and common ratio r = 2

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:
`a_n= a_1r^(n-1)`

We are given: third term = 6 i,e

`a_3=a_1r(3-1)\\6=a_1r^2----eq(1)`

Seventh term = 96

`a_7=a_1r(7-1)\\96=a_1r^6----eq(2)`

Dividing eq(2) and eq(3)

`(96)/(6)=(a_1r^6)/(a1r2)\\16=(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}`

So, Common Ratio r = 2

Finding First term using eq(1)

`a_3=a_1r^2\\6=a_1(2)^2\\6=a_1(4)\\a_1= (6)/(4)\\a_1= (3)/(2)`

So, First term a₁ = 3/2 and common ratio r = 2