In a G.P, the 5th term exceeds the 4th term by 24, and the 4th term exceeds the 3rd term by 8, find the common ratio and the first term.
Answers
- The 5th term exceeds the 4th term by 24.
- The 4th term exceeds the 3rd term by 8.
- Formula of nth term of G.P.
⇒ Tₙ = arⁿ⁻¹.
- The 5th term exceeds the 4th term by 24.
⇒ T₅ = T₄ + 24.
⇒ T₅ - T₄ = 24.
⇒ ar⁴ - ar³ = 24.
⇒ ar³(r - 1) = 24. ____(1).
- The 4th term exceeds the 3rd term by 8.
⇒ T₄ = T₃ + 8.
⇒ T₄ - T₃ = 8.
⇒ ar³ - ar² = 8.
⇒ ar²(r - 1) = 8. ____(2).
- Divide equation (1) by (2), we get.
⇒ ar³(r - 1) = 24. ____(3).
⇒ ar²(r - 1) = 8. ____(4).
We get,
⇒ r = 3.
Put the value of r = 3 in the equation (3), we get.
⇒ ar³(r - 1) = 24.
⇒ a(3)³(3 - 1) = 24.
⇒ a(27)(2) = 24.
⇒ 54a = 24.
⇒ a = 24/54.
⇒ a = 4/9.
First term = a = 4/9.
Common ratio = r = 3.
The 5th term exceeds the 4th term by 24.
The 4th term exceeds the 3rd term by 8.
As we know that,
Formula of nth term of G.P.
⇒ Tₙ = arⁿ⁻¹.
The 5th term exceeds the 4th term by 24.
⇒ T₅ = T₄ + 24.
⇒ T₅ - T₄ = 24.
⇒ ar⁴ - ar³ = 24.
⇒ ar³(r - 1) = 24. - - - - - (1).
The 4th term exceeds the 3rd term by 8.
⇒ T₄ = T₃ + 8.
⇒ T₄ - T₃ = 8.
⇒ ar³ - ar² = 8.
⇒ ar²(r - 1) = 8. - - - - - (2).
Divide equation (1) by (2), we get.
⇒ ar³(r - 1) = 24. - - - - - (1).
⇒ ar²(r - 1) = 8. - - - - - (2).
We get,
⇒ r = 3.
Put the value of r = 3 in the equation (1), we get.
⇒ ar³(r - 1) = 24.
⇒ a(3)³(3 - 1) = 24.
⇒ a(27)(2) = 24.
⇒ 54a = 24.
⇒ a = 24/54.
⇒ a = 4/9.
First term = a = 4/9.
Common ratio = r = 3.