MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1966 AMC 12/AHSME
36
Identity with Series
Identity with Series
Source: 1966 AHSME #36
August 31, 2011
AMC
Problem Statement
Let
(
1
+
x
+
x
2
)
n
=
a
0
+
a
1
x
+
a
2
x
2
+
.
.
.
+
a
2
n
x
2
n
(1+x+x^2)^n=a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}
(
1
+
x
+
x
2
)
n
=
a
0
+
a
1
x
+
a
2
x
2
+
...
+
a
2
n
x
2
n
be an identity in
x
x
x
. If we lt
s
=
a
0
+
a
2
+
a
4
+
.
.
.
+
a
2
n
s=a_0+a_2+a_4+...+a_{2n}
s
=
a
0
+
a
2
+
a
4
+
...
+
a
2
n
, then
s
s
s
equals:
(A)
2
n
(B)
2
n
+
1
(C)
3
n
−
1
2
(D)
3
n
2
(E)
3
n
+
1
2
\text{(A)}\ 2^n\qquad \text{(B)}\ 2^n+1\qquad \text{(C)}\ \dfrac{3^n-1}{2}\qquad \text{(D)}\ \dfrac{3^n}{2}\qquad \text{(E)}\ \dfrac{3^n+1}{2}
(A)
2
n
(B)
2
n
+
1
(C)
2
3
n
−
1
(D)
2
3
n
(E)
2
3
n
+
1
Back to Problems
View on AoPS