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3
I 3
I 3
Source:
May 25, 2007
floor function
induction
Problem Statement
Prove that for any positive integer
n
n
n
,
⌊
n
+
1
2
⌋
+
⌊
n
+
2
4
⌋
+
⌊
n
+
4
8
⌋
+
⌊
n
+
8
16
⌋
+
⋯
=
n
.
\left\lfloor \frac{n+1}{2}\right\rfloor+\left\lfloor \frac{n+2}{4}\right\rfloor+\left\lfloor \frac{n+4}{8}\right\rfloor+\left\lfloor \frac{n+8}{16}\right\rfloor+\cdots = n.
⌊
2
n
+
1
⌋
+
⌊
4
n
+
2
⌋
+
⌊
8
n
+
4
⌋
+
⌊
16
n
+
8
⌋
+
⋯
=
n
.
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