show that all the values of i^i are real and in GP
Answers
Step-by-step explanation:
Answer
Let the three terms in GP be x,xr, xr
2
x+xr+xr
2
=aS
x
2
+x
2
r
2
+x
2
r
4
=S
2
Therefore,
1+r
2
+r
4
(1+r+r
2
)
2
=a
2
(1+r+r
2
)×(1−r+r
2
)
(1+r+r
2
)
2
=a
2
(1−r+r
2
)
(1+r+r
2
)
=a
2
(1−a
2
)r
2
+(1+a
2
)r+(1−a
2
)=0
Since r is real, (1+a
2
)
2
−4(1−a
2
)
2
≥0
Let a
2
=t
(1+t)
2
−4(1−t)
2
≥0
(3t−1)(t−3)≤0
From the equation, we get
3
1
≥a
2
≤3
But for a
2
=1, r=0 and GP will not be defined.
Hence,
3
1
≥a
2
<1 and 1>a
2
≤3
Answer:
Let the three terms in GP be x,xr, xr
2
x+xr+xr
2
=aS
x
2
+x
2
r
2
+x
2
r
4
=S
2
Therefore,
1+r
2
+r
4
(1+r+r
2
)
2
=a
2
(1+r+r
2
)×(1−r+r
2
)
(1+r+r
2
)
2
=a
2
(1−r+r
2
)
(1+r+r
2
)
=a
2
(1−a
2
)r
2
+(1+a
2
)r+(1−a
2
)=0
Since r is real, (1+a
2
)
2
−4(1−a
2
)
2
≥0
Let a
2
=t
(1+t)
2
−4(1−t)
2
≥0
(3t−1)(t−3)≤0
From the equation, we get
3
1
≥a
2
≤3
But for a
2
=1, r=0 and GP will not be defined.
Hence,
3
1
≥a
2
<1 and 1>a
2
≤3