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Simon Marais Mathematical Competition
2017 Simon Marais Mathematical Competition
A2
A2
Part of
2017 Simon Marais Mathematical Competition
Problems
(1)
sequence with reciprocal of sum of squares
Source: Simon Marais 2017 A2
4/29/2021
Let
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\ldots
a
1
,
a
2
,
a
3
,
…
be the sequence of real numbers defined by
a
1
=
1
a_1=1
a
1
=
1
and
a
m
=
1
a
1
2
+
a
2
2
+
…
+
a
m
−
1
2
for
m
≥
2.
a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.
a
m
=
a
1
2
+
a
2
2
+
…
+
a
m
−
1
2
1
for
m
≥
2.
Determine whether there exists a positive integer
N
N
N
such that
a
1
+
a
2
+
…
+
a
N
>
201
7
2017
.
a_1+a_2+\ldots+a_N>2017^{2017}.
a
1
+
a
2
+
…
+
a
N
>
201
7
2017
.
algebra
Sequences