1. Find term a20 of the Padovan sequence as de?ned in the notes. 2. Suppose we s
ID: 2979860 • Letter: 1
Question
1. Find term a20 of the Padovan sequence as de?ned in the notes. 2. Suppose we specify that x= 1 and for n > 1 we let
Xn= 1 +max{xm : m < n} if n is odd
2min{xm : m < n} if n is even
Specify x11
3. Analyze the following recursive sequences. In all
cases these have the form an+1 = f(an) and you
should find f(x) and compute the required term.
The first one is an example.
EXAMPLE 3, 6, 12, 24, 48, . . . ... so a1 = 3and
an+1 = f(an). What is a7? Answer is 192
(A) 9,?18, 36,?72, . . . so b1 = 9 and
bn+1 = g(bn). What is b7?
(B) 1, 3, 7, 15, 31, 63, . . . so c1 = 1 and
cn+1 = h(cn). What is c10?
(C) 66, 33, 100, 50, 25, 76, . . . so d1 = 66 and
dn+1 = k(dn). What is d10?
(Hint:See page 34.)
Explanation / Answer
1. Padovan
a1=a2=a3=1 (by rule)
a4= a2+a1=2
a5=a3+a2=2
a6=a4+a3=2+1=3
a7=a5+a4=2+2=4
a8=a6+a5=5
a9=a7+a6=7
keep going on as above and get the series.
2.)
x1=1
min{xm : m < n} is always equal to 1. since 1 is the minimum value possible.
thus all even terms will be equal to 2 .
x2= 2min{xm : m < n} =2
x3=1 +max{xm : m < n}= 1 + 2= 3 (since x2=2 is max term before x3)
x4= x6=x8=x10= 2
x5=1 +max{xm : m < n} = 1+ 3= 4 (since max term is x3=3 before the 5th term)
x7=1 +max{xm : m < n}=1 + 4 =5
x9=1 +max{xm : m < n}= 1 + 5 = 6
x11 =1 +max{xm : m < n} = 1+ 6 = 7
thus x11 = 7
3.)
A)576
next term arrives by multiplying -2
i.e
an+1= -2*an
example; a2=-2*a1=-2 *9=-18; a3= -18*-2 = 72; a4= 72 *-2= -144
B) 1023
an+1 = 2*an +1
example: a2= 2a1 +1 = 2*1 +1=3; a3 = 2*3 +1 =7; a4= 2*7+1= 15
C)29
an+1 =an/2, if an is an even number
an+1 = an+an-1 +1 , if an is odd
example : a2=66/2=33; a3= 66+33+1 = 100; a4= 100/2 = 50; a5=50/2 = 25;
a6= 25 + 50 +1 = 76
Hope you got how the series is formed and the hence the numbers
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