A G.P consists of an even number of terms. If the sum of all terms is 5 times the sum of terms occupying odd places, then find its common ratio

Solution:

Let the G.P be T₁, T₂, T₃, T₄, ...., T₂ₙ.

According to the question,

⇒ T₁ + T₂ + T₃ + T₄ + .... + T₂ₙ = 5[T₁ + T₃ + T₅ + .... + T₂ₙ ₋ ₁]

⇒ T₁ + T₂ + T₃ + T₄ + .... + T₂ₙ = 5[T₁ + T₃ + T₅ + .... + T₂ₙ ₋ ₁] = 0

⇒ T₂ + T₄ + .... + T₂ₙ = 4[T₁ + T₃ + .... + T₂ₙ ₋ ₁]

Let the G.P be a, ar, ar², .... i.e., a, ar, ar², .... = T₁, T₂, T₃, T₄, ....

ar (rⁿ - 1)/(r - 1) = 4 x a (rⁿ - 1)/(r - 1)

⇒ ar = 4a

⇒ r = 4

Thus, the common ratio of the G.P is 4

NCERT Solutions Class 11 Maths Chapter 9 Exercise ME Question 11

A G.P consists of an even number of terms. If the sum of all terms is 5 times the sum of terms occupying odd places, then find its common ratio

Summary:

If the sum of all terms of a GP of even number of terms is 5 times the sum of terms occupying odd places, then its common ratio is 4