Question

In a GP of positive terms, if any terms is equal to the sum of next two terms find the common ratio of GP..​

Answer 1

Answer:

r = (-1 + √5)/2

Step-by-step explanation:

Let a, ar, ar² be three positive terms of G.P.

According to given condition,

⇒ a = ar + ar²

⇒ 1 = (ar + ar²)/a

⇒ 1 = a(r + r²)/a

⇒ 1 = r + r²

⇒ r² + r - 1 = 0

∴ D = b² - 4ac

      = (1)² - 4(1)(-1)

      = 5

Now,

r = -b ± √D/2a

  = -(1) ± √5/2

  = (-1 ± √5)/2

Since, r can not be negative.

∴ r = (-1 + √5)/2

Hope it helps!

Answer 2

Answer:

r = (-1 + √5)/2

Step-by-step explanation:

Let a, ar, ar² be three positive terms of G.P.

According to given condition,

⇒ a = ar + ar²

⇒ 1 = (ar + ar²)/a

⇒ 1 = a(r + r²)/a

⇒ 1 = r + r²

⇒ r² + r - 1 = 0

∴ D = b² - 4ac

      = (1)² - 4(1)(-1)

      = 5

Now,

r = -b ± √D/2a

  = -(1) ± √5/2

  = (-1 ± √5)/2

Since, r can not be negative.

∴ r = (-1 + √5)/2