What are the first 4 terms of the G.P.whose nth term is ....
Answers
Step-by-step explanation:
Given that, there are 4n+1 terms in a sequence of which first 2n+1 are in Arithmetic Progression and last 2n+1 are in Geometric Progression the common difference of Arithmetic Progression is 2 and common ratio of Geometric Progression is
2
1
.
Let, a be the first term of AP.
⇒ first term of GP is a+(2n+1−1)d=a+4n
Middle terms of AP and GP are equal.
⇒a+2n=
2
n
a+4n
-----(1)
But, Middle term of the whole sequence is T
m
which is sum of infinite GeometricProgression whose sum of first Two terms is (
4
5
)
2
n and ratio of these terms is
16
9
let A be the first term of infinite geometric series.
⇒A(1+r)=
16
25
n
⇒A=n
⇒T
m
=a+4n=
1−r
A
=
7
16n
-----(2)
from (1) and (2)
7
16n
−2n=
7.2
n
16n
⇒n=3
⇒T
m
=a+4n=
7
16n
=
7
48
.
∴ First term of the sequence a=
7
48
−12=
7
−36