Download Sequences/Series

2009 – 2010 Log1 Contest Round 2
Theta Sequences and Series
Name: __________________
4 points each
1
What is the next term in the arithmetic sequence: 3, 7, 11… ?
2
What is the sum of the first 10 terms of the arithmetic sequence
2, 5, 8…
If a 1 = 3x + 17 and a 4 = 73 − 4x are elements of an arithmetic sequence, then what
3
4
5
6
7
8
is the common difference in terms of x?
1
Let an = . What is a 1 + a 2 + a 3 expressed as an improper fraction?
n
Evaluate:
6 + 6 + 6 + 6 + ...
5 points each
Let Sn be the sum of the first n positive perfect squares. What is S3 + S 5 + S 7 ?
What is the positive difference between the harmonic mean and the arithmetic mean
of 4 and 8?
What is the sum of the following infinite geometric series: 54 + 36 + 24 + ... ?
9
A square is inscribed in a circle of diameter 2. Another circle is then inscribed in the
square and then a square is inscribed into that circle. If this process continues
infinitely, what is the total area of the squares?
10
If 0.84... is expressed as a reduced fraction
n
, then what is n+m?
m
6 points each
11
A sequence is defined recursively by:
a0 = 4, a1 = 1, an +2 =
12
13
2an +1 + an
. What is a 4 , expressed as an improper
3
fraction?
Evaluate the sum of the following infinite series:
1 3
1
3
−
+
−
+ ...
4 16 64 256
Let an be an arithmetic sequence such that a 1 , a 4 , and a 13 creates a geometric
sequence. If a14 = 58 , then what is the common difference of an ?
14
15
10

2


Find the sum and express your answer as an improper fraction: ∑ 
2
n =2  n − 1 
A set of lines in the first quadrant has a special quality. The x-intercepts of the lines
form an arithmetic sequence with common difference of 1 and the y-intercepts of
the lines form a geometric sequence with a common ratio of 2. What is the point of
intersection between the third line of this set and the line y = x , if the first line
has a standard equation of x + y = 1 ?
2009 – 2010 Log1 Contest Round 2
Alpha Sequences and Series
Name: __________________
1
2
3
4 points each
What is the next term in the arithmetic sequence: 3, 7, 11… ?
What is the sum of the first 10 terms of the arithmetic sequence
2, 5, 8…
If a 1 = 3x + 17 and a 4 = 73 − 4x are elements of an arithmetic sequence, then what
4
is the common difference in terms of x?
1
Let an = . What is a 1 + a 2 + a 3 expressed as an improper fraction?
5
Evaluate:
6
5 points each
Let Sn be the sum of the first n positive perfect squares. What is S3 + S 5 + S 7 ?
7
8
n
13
13
12 +
13
12 +
13
12 +
12 + ...
What is the positive difference between the harmonic mean and the arithmetic mean
of 4 and 8?
What is the sum of the following infinite geometric series: 54 + 36 + 24 + ... ?
9
A square is inscribed in a circle of diameter 2. Another circle is then inscribed in the
square and then a square is inscribed into that circle. If this process continues
infinitely, what is the total area of the squares?
10
If 0.89772 … is expressed as a reduced fraction
11
A sequence is defined recursively by:
n
, then what is n+m?
m
6 points each
a0 = 4, a1 = 1, an +2 =
12
13
2an +1 + an
. What is a 4 , expressed as an improper
3
fraction?
Evaluate the sum of the following infinite series:
1 3
1
3
−
+
−
+ ...
4 16 64 256
Let an be an arithmetic sequence such that a 1 , a 4 , and a 13 creates a geometric
sequence. If a14 = 58 , then what is the common difference of an ?
14
15
10

2


Find the sum and express your answer as an improper fraction: ∑ 
2
n =2  n − 1 
Evaluate:
180°
2
∑ sin x .
x = 0°
Name: __________________
1
2
3
2009 – 2010 Log1 Contest Round 2
Mu Sequences and Series
4 points each
What is the next term in the arithmetic sequence: 3, 7, 11… ?
What is the sum of the first 10 terms of the arithmetic sequence
2, 5, 8…
If a 1 = 3x + 17 and a 4 = 73 − 4x are elements of an arithmetic sequence, then what
is the common difference in terms of x?
4
5
6
7
8
Let an =
1
n
. What is
4
an expressed as an improper fraction?
∑
n
=1
13
Evaluate:
13
12 +
13
12 +
13
12 +
12 + ...
5 points each
Let Sn be the sum of the first n positive perfect squares. What is S3 + S 5 + S 7 ?
What is the positive difference between the harmonic mean and the arithmetic mean
of 4 and 8?
What is the sum of the following infinite geometric series: 54 + 36 + 24 + ... ?
9
An equilateral triangle is inscribed in a circle of diameter 2. Another circle is then
inscribed in the triangle and then an equilateral triangle is inscribed into that circle.
If this process continues infinitely, what is the total area of the triangles?
10
If 0.89772 … is expressed as a reduced fraction
11
A sequence is defined recursively by:
n
, then what is n+m?
m
6 points each
a0 = 4, a1 = 1, an +2 =
12
13
2an +1 + an
. What is lim an , expressed as an improper
3
n →∞
fraction?
Evaluate the sum of the following infinite series:
1 3
1
3
−
+
−
+ ...
4 16 64 256
Let an be an arithmetic sequence such that a 1 , a 4 , and a 13 creates a geometric
sequence. If a14 = 58 , then what is the common difference of an ?
14
15
10

2


Find the sum and express your answer as an improper fraction: ∑ 
2
n =2  n − 1 
Evaluate:
180°
2
∑ sin x .
x = 0°
2009 – 2010 Log1 Contest Round 2
Sequences and Series Answers
Theta Answers
Alpha Answers
Mu Answers
1
15
1
15
1
15
2
155
2
155
2
155
3
56 − 7 x
3
3
56 − 7 x
3
3
56 − 7 x
3
4
11
6
4
11
6
4
25
12
5
3
5
1
5
1
6
209
6
209
6
209
7
2
3
7
2
3
7
2
3
8
162
8
162
8
162
9
4
9
4
9
3
10
83
10
167
10
167
11
16
9
11
16
9
11
7
4
12
1
15
12
1
15
12
1
15
13
4
13
4
13
4
14
72
55
14
72
55
14
72
55
15
 12 12 
 , 
7 7 
15
90
15
90
2009 – 2010 Log1 Contest Round 2
Sequences and Series Solutions
Mu
1
2
Al
1
2
Th
1
2
3
3
3
Solution
Common difference = 4 ∴ 11 + 4 = 15
t1 = 2
t10 = 2 + 9*3 = 29
2 + 29
10(
) = 155
2
Common difference = r
3 x + 17 + 3r = 73 − 4 x
56 − 7 x
3
1 1 1 11
+ + =
1 2 3 6
1 1 1 1 25
+ + + =
1 2 3 4 12
r=
4
4
4
5
x = 6 + 6 + 6 + 6 + ...
x = 6+ x
x2 = 6 + x
x2 − x − 6 = 0
x = −2,3
x=3
5
13
13
12 +
13
12 +
13
12 +
12 + ...
13
x=
12 + x
x( x + 12) = 13
5
x=
x 2 + 12 x − 13 = 0
x = 1,−13
6
6
6
x =1
S3 = (1 + 4 + 9) = 14
S5 = (1 + 4 + 9 + 16 + 25) = 55
S7 = (1 + 4 + 9 + 16 + 25 + 36 + 49) = 140
S3 + S5 + S7 = 209
2ab
a+b
a+b
Arithmetic mean =
2
2( 4)(8) 4 + 8 2
−
=
4+8
2
3
7
7
7
Harmonic mean =
8
8
8
r=
36 2
=
54 3
t1
54
=
= 162
2
1−r
1−
3
9
9
The diameter of the circle is the diagonal of the square. The first square has area
then of (2)(2)/2 = 2. The diameter of the second circle is the side length of the first
2 so that the area of the second square is 1. The area of all the
1 1
squares is: 2 + 1 + + + L = 4
2 4
square equals
9
The center of the circle is the centroid of the triangle which means the diameter of
the second circle is half the diameter of the first circle and the area will be ¼ of the
first. The first triangle will have area
3 3
4
3 3
TA = 4 = 3
1
1−
4
t1 =
10
10
10
n
= 0.84
m
10n
76
= 8 94 =
m
9
n 76 38
=
=
m 90 45
n + m = 38 + 45 = 83
n
= 0.89772
m
72
n
1000 = 897.72 = 897 +
99
m
= 897 +
11000
8
11
n
= 9875
m
9875 79
n
=
=
m 11000 88
11
11
11
12
79 + 88 = 167
2(1) + 4
a2 =
=2
3
2(2) + 1 5
a3 =
=
3
3
2(5 3) + 2 16
a4 =
=
3
9
The sequence may be written:
12
12
3an +2 − 2an +1 − an = 0 . The solution will be of the form ax1n + bx2n where
1
x1 , x2 solve 3x 2 − 2x − 1 = 0 . The solution a ( − )n + b (1)n can be solved using the
3
9 1 n 7 n
first two values. The solution is: an = ( − ) + (1) which goes to 7/4.
4
4 3
Group the series two terms at a time and simplify:
1
1
1
16 = 1  16  = 1
+
+L =
16 256
1 − 116 16  15  15
13
13
13
Let x be the common difference of an:
a14 = 58
a13 = 58 − x
a 4 = 58 − 10x
a1 = 58 − 13x
a13 a 4
58 − x
58 − 10x
=
→
=
a4
a1
58 − 10x 58 − 13x
Cross-multiplying and solving for x yields x = 4 .
14
14
14
15
10
 10  1
1 
 = ∑ 
−

2
n =2  n − 1  n =2  n − 1 n + 1 
1 1 1  1 1 1 1
1  1
1
1

= 1 −  +  −  +  −  +  −  + L +  −
+ − 
3 2 4  3 5  4 6
 8 10   9 11 

1
1
1
72
=1+ −
−
=
2 10 11 55

∑ 
2
line
x +y =1
x-int.
(1,0)
#2
(2,0)
#3
(3,0)
4
Slope of #3 = −
3
4
∴y = − x + 4
3
y-int.
(0,1)
(0,2)
(0,4)
 12 12 
Solving the system for line #3 and y = x yields the intersection point of  ,  .
7 7
15
15
Use the identity: cos2 (180 − x ) = sin2 (x ) and the sum becomes:
89°
(
)
2
2
2
∑ sin x + cos x + sin 90° = 90
x = 0°