Respuesta :
Answer:
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 = [tex]\frac{1+\sqrt{5} }{2}[/tex] , [tex]\frac{1-\sqrt{5} }{2}[/tex]
(roots of ax²+bx+c are [tex]\frac{-b+\sqrt{b^{2} -4ac} }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^{2} -4ac} }{2a}[/tex])
and it is given, increasing geometric progression
⇒ r > 0.
⇒ r = [tex]\frac{1+\sqrt{5} }{2}[/tex].
Now, nth term in geometric progression is arⁿ⁻¹.
⇒ 7th term = ar⁷⁻¹ = ar⁶.
= (9-4√5)([tex]\frac{1+\sqrt{5} }{2}[/tex])⁶
= (0.05572809)(17.94427191) = 1
⇒ 7th term = 1.
