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Mathematics
Problem
Consider the infinite geometric series
2
k
+
2
k
(
k
−
1
k
+
1
)
+
2
k
(
k
−
1
k
+
1
)
2
+
⋯
.
2k+2k\left(\dfrac{k-1}{k+1}\right)+2k\left(\dfrac{k-1}{k+1}\right)^2+\cdots.
2
k
+
2
k
(
k
+
1
k
−
1
)
+
2
k
(
k
+
1
k
−
1
)
2
+
⋯
.
If its sum to infinity equals
8
8
8
, then the value of
k
k
k
is a)
k
=
7
k = 7
k
=
7
b)
k
=
2
k = 2
k
=
2
c)
k
=
−
1
+
33
2
k = \displaystyle \frac{-1+\sqrt{33}}{2}
k
=
2
−
1
+
33
d)
k
=
1
+
33
2
k = \displaystyle \frac{1+\sqrt{33}}{2}
k
=
2
1
+
33
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