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Solved Problems on Introduction to
Sequences
Solved Problems on Introduction to Sequences by Mika Seppälä
Sequences
DefinitionDefinition
A sequence (an)=(a1, a2, a3, …) is a rule that assigns number an to every positive integer n.
Solved Problems on Introduction to Sequences by Mika Seppälä
1
SEQUENCES
1,
1
2,1
6,
1
24,K
⎛
⎝⎜⎞
⎠⎟
(3., 3.1, 3.14, 3.141, 3.1415,…)2
3 2,
5
4,10
9,17
16,K
⎛
⎝⎜⎞
⎠⎟
Find the general rule defining the following sequences.
Solved Problems on Introduction to Sequences by Mika Seppälä
4
SEQUENCES
s1=n
Let n be a positive integer. Define the sequence (sk) by
sk+1=3s
k+1, if s
k is odd.
Let n = 3, compute (sk).
Solved Problems on Introduction to Sequences by Mika Seppälä
1
SEQUENCES
Solution The denominators are
Answer
The general nth term is .
1,
1
2,1
6,
1
24,K
⎛
⎝⎜⎞
⎠⎟ = a
n( )
1, 2, 6, 24,…= 1, 2∙1, 3∙2∙1, 4∙3∙2∙1,…
a
n=
1n!
Solved Problems on Introduction to Sequences by Mika Seppälä
2
SEQUENCES
Solution Remember
π =3.1415926535897932846246….
Answer
(3., 3.1, 3.14, 3.141, 3.1415,…)
The general nth term is n first numbers in the decimal point expansion of π
Solved Problems on Introduction to Sequences by Mika Seppälä
3
SEQUENCES
Solution Clearly
Answer
= a
n( )
for some f.
a
n=
n2 +1n2
=1 +1n2
.
2,
5
4,10
9,17
16,K
⎛
⎝⎜⎞
⎠⎟
an=
f n( ) +1
f n( )f(1) = 1,
f(2) = 22, f(3) = 9 = 32, f(4) = 16 = 42.
Solved Problems on Introduction to Sequences by Mika Seppälä
4
SEQUENCES
s1=n
Let n be a positive integer. Define the sequence (sk) by
sk+1=3s
k+1, if s
k is odd.
Let n = 3, compute (sk).
Solved Problems on Introduction to Sequences by Mika Seppälä
4
SEQUENCES
s1=3
sk+1=3s
k+1, if s
k is odd.
Solution s1=3 s2
=3 ⋅s1+1 =10
s
3=
s2
2 =5
s4=3 ⋅s
3+1 =16 s5
=8 s6=4 s7
=2 s8=1
Answer