Answer:
Answer
It can be (1+x)
n
only as (1−x)
n
terms are +ve/−ve alternatively hence can't be in A.P.,
General term, T
rH
=
n
C
r
⋅x
r
⇒T
5
=
n
C
4
⋅x
4
,T
6
=
n
C
5
⋅x
5
,T
7
=
n
C
6
⋅x
6
Coefficients are in A.P.
⇒2
n
C
5
=
n
C
4
+
n
C
6
⇒
5!(n−5)!
2n!
=
4!(n−4)!
n!
+
6!(n−6)!
n!
⇒
5(n−5)
2
=
(n−4)(n−5)
1
+
6×5
1
⇒2(n−4)=5+
6
1
(n−4)(n−5)
⇒12(n−4)=30+n
2
−9n+20
12n−48=n
2
−9n+50
⇒n
n
−21n+98=0
⇒n
2
−14n−7n+98=0
⇒n(n−14)−7(n−14)=0
⇒(n−7)(n−14)=0
n=7 or n=14.