MathDB
Problems
Contests
International Contests
Tournament Of Towns
1985 Tournament Of Towns
(102) 6
(102) 6
Part of
1985 Tournament Of Towns
Problems
(1)
TOT 102 1985 Autumn J6 x_{k+1} =x^2_k+x_k, sum 1/{x_i+1}
Source:
8/26/2019
The numerical sequence
x
1
,
x
2
,
.
.
x_1 , x_2 ,..
x
1
,
x
2
,
..
satisfies
x
1
=
1
2
x_1 = \frac12
x
1
=
2
1
and
x
k
+
1
=
x
k
2
+
x
k
x_{k+1} =x^2_k+x_k
x
k
+
1
=
x
k
2
+
x
k
for all natural integers
k
k
k
. Find the integer part of the sum
1
x
1
+
1
+
1
x
2
+
1
+
.
.
.
+
1
x
100
+
1
\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}
x
1
+
1
1
+
x
2
+
1
1
+
...
+
x
100
+
1
1
{A. Andjans, Riga)
Sequence
Sum
Integer Part
recurrence relation
algebra