Find the seventh term of an increasing geometric progression if the first term is equal to 9−4sqrt5 and each term (starting with the second) is equal to the difference of the term following it and the term preceding it.

7th term = 1.

Step-by-step explanation:

Given that, first term of increasing geometric progression is 9-4√5.

each term (starting with the second) is equal to the difference of the term following it and the term preceding it.

let first term of geometric progression be a and the increasing ratio be r.

⇒ The geometric progression is   a , ar , ar² , ar³, ....... so on.

Given, each term (starting with the second) is equal to the difference of the term following it and the term preceding it.

⇒ second term = (third term - first term)

⇒ ar = (ar² - a)

⇒ r = r² - 1

⇒ r² - r -1 =0

⇒ roots of this equation is r =  ,

 (roots of ax²+bx+c are and )

and it is given, increasing geometric progression

⇒ r > 0.

⇒ r = .

Now, nth term in geometric progression is arⁿ⁻¹.

⇒ 7th term = ar⁷⁻¹ = ar⁶.

     = (9-4√5)()⁶

     = (0.05572809)(17.94427191)  =  1

⇒ 7th term = 1.

x= -3

step-by-step explanation:

step 1: simplify brackets

4 - x - 1 = 6

step 2: simpilfy

3 - x = 6

step 3: subtract 3 from both sides

-x = 6 - 3

step 4: simpilfy 6 - 3

-x = 3

step 5: multiply both sides by -1

x = -3