Question

Find the 4th term from the end of the g.p 2/51, 2/27, 2/9,------,54 ?​

Answer 1

Correct Question:

Find the 4th term from the end of the GP 2/81, 2/27, 2/9, ........ , 54.

Answer:

2

Step-by-step explanation:

Given :

2/81, 2/27, 2/9, ....... , 54 are in GP

First term of the GP ( a ) = 2/81

Common ratio of GP ( r ) = a₂ / a₁ = 2/27 ÷ 2/81 = 2/27 × 81/2 = 3

Let last term of the GP aₙ = 54

Using nth term of GP  formula

⇒ aₙ = arⁿ ⁻ ¹

$\Rightarrow \sf 54 = \dfrac{2}{81}\times3^{n-1} \\\\\\\Rightarrow\sf27=\dfrac{1}{81} \times 3^{n-1} \\\\\\\Rightarrow\sf27=\dfrac{3^{n-1} }{3^4} \\\\\\\Rightarrow\sf 3^3= 3^{n-1-4} \\\\\\\Rightarrow\sf3^3= 3^{n-5}$

Since bases are equal we can equate powers

⇒ 3 = n - 5

⇒ 3 + 5 = n

⇒ n = 8

∴ GP has 8 terms. 4th term from the end of the GP will be the 5th term of the GP.

Again using nth term of GP formula

⇒ aₙ = arⁿ ⁻ ¹

⇒ 5th term = a₅ = ar⁵ ⁻ ¹

⇒ a₅ = ar⁴

⇒ a₅ = 2/81 × 3⁴

⇒ a₅ = 2/81 ×  81 = 2

Therefore 4th term from the end of the GP is 2.