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Eotvos Mathematical Competition (Hungary)
1923 Eotvos Mathematical Competition
2
2
Part of
1923 Eotvos Mathematical Competition
Problems
(1)
sum B(n+1, k)= 2^n S_n
Source: Eotvos 1923 p2
9/10/2024
If
s
n
=
1
+
q
+
q
2
+
.
.
.
+
q
n
s_n = 1 + q + q^2 +... + q^n
s
n
=
1
+
q
+
q
2
+
...
+
q
n
and
S
n
=
1
+
1
+
q
2
+
(
1
+
q
2
)
2
+
.
.
.
+
(
1
+
q
2
)
n
,
S_n = 1 +\frac{1 + q}{2}+ \left( \frac{1 + q}{2}\right)^2 +... + \left( \frac{1 + q}{2}\right)^n,
S
n
=
1
+
2
1
+
q
+
(
2
1
+
q
)
2
+
...
+
(
2
1
+
q
)
n
,
prove that
(
n
+
1
1
)
+
(
n
+
1
2
)
s
1
+
(
n
+
1
3
)
s
2
+
.
.
.
+
(
n
+
1
n
+
1
)
s
n
=
2
n
S
n
{n + 1 \choose 1}+{n + 1 \choose 2} s_1 + {n + 1 \choose 3} s_2 + ... + {n + 1 \choose n + 1} s_n = 2^nS_n
(
1
n
+
1
)
+
(
2
n
+
1
)
s
1
+
(
3
n
+
1
)
s
2
+
...
+
(
n
+
1
n
+
1
)
s
n
=
2
n
S
n
binomial coefficients
Binomial
algebra
Sum