The nth term of a GP is 128 and the sum of its n terms is 255 If its common ratio is 2 then find first term

Answers

Question :

The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find first term .
Submitted on 8/7/2024 | Answered by Vandana Rana

Answer :

The answer to this question will be as following:


 a_n
 
 =
 128
 ,

 
 S_n
 
 =
 255

 and
 r
 =
 2

 
 a_n
 
 =
 128

 
 ∴
 a
 r^n-1
 
 =
 128

 
 ⇒
 a2^n-1
 
 
 =
 128
 

 a2ⁿ=256
 
 .......... (i)

 
 Also,
 S_n
 
 =
 255

 
 ⇒
 a
 (
 rⁿ
 
 −
 1)/(
 r
 −
 1
 )
 =
 255

 
 ⇒
 a
 (
 2ⁿ
 
 −
 1
 )/(2-1)
 =
 255

 
 ⇒
 a2ⁿ- a =255

 (
 256
 a
 −
 1
 )
 =
 255
 [Using (i)]

 
 ⇒
 256
 −
 a
 =
 255

 
 ⇒
 a
 =
 256
 −
 255

 
 ⇒
 a
 =
 1