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Mathematics
Problem
Suppose
u
1
,
u
2
,
…
,
u
n
u_1, u_2, \dots, u_n
u
1
,
u
2
,
…
,
u
n
is a geometric sequence with common ratio
r
r
r
. Show that
(
u
1
+
u
2
)
2
+
(
u
2
+
u
3
)
2
+
(
u
3
+
u
4
)
2
+
⋯
+
(
u
n
−
1
+
u
n
)
2
=
2
u
1
2
(
r
2
n
−
1
−
1
)
r
−
1
−
(
u
1
2
+
u
n
2
)
.
(u_1 + u_2)^2 + (u_2 + u_3)^2 + (u_3 + u_4)^2 + \dots + (u_{n-1} + u_n)^2 = \frac{2u_1^2(r^{2n-1} - 1)}{r - 1} - (u_1^2 + u_n^2).
(
u
1
+
u
2
)
2
+
(
u
2
+
u
3
)
2
+
(
u
3
+
u
4
)
2
+
⋯
+
(
u
n
−
1
+
u
n
)
2
=
r
−
1
2
u
1
2
(
r
2
n
−
1
−
1
)
−
(
u
1
2
+
u
n
2
)
.
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