Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions.
Solution:
Step1
To find
We have to find the first five terms of the sequence defined by each of these recurrence relations and initial conditions.
Step2
Sequence:-An arrangement is a specified collection of objects in which repetitions are permitted.
Recurrence relations:-The technique for finding the terms of an arrangement in a recursive way is called recurrence relations.
a.
First term
When n=1
Second term
=
= 6.2
= 12
When n=2
Third term
=
= 6.12
= 72
When n=3
Fourth term
=
= 6.72
= 432
When n=4
Fifth term
=
= 6.432
= 2592
Therefore, the first five terms of the sequence defined by each of these recurrence relations and initial conditions is 2,12,72,432,2592.
Step3
b.
First term
When n=2
Second term
=
=
=4
When n=3
Third term
=
=
=16
When n=4
Fourth term
=
=
=256
When n=5
Fifth term
=
=
=65536
Therefore, the first five terms of the sequence defined by each of these recurrence relations and initial conditions is 2,4,16,256,65536.
Step4
c.
First term
Second term
When n=2
Third term
=
=2+3.1
=2+3
=5
When n=3
Fourth term
=
=5+3.2
=5+6
=11
When n=4
Fifth term
=
=11+3.5
=11+15
=26
Therefore, the first five terms of the sequence defined by each of these recurrence relations and initial conditions is 1,2,5,11,26.
Step5
d.
First term
Second term
When n=2
Third term
=
=2.1+4.1
= 2+4
=6
When n=3
Fourth term
=
=3.6+9.1
= 18+9
=27
When n=4
Fifth term
=
=4.27+16.6
= 108+96
=204
Therefore, the first five terms of the sequence defined by each of these recurrence relations and initial conditions is 1,1,6,27,204.
Step6
e.
First term
Second term
Third term
When n=3
Fourth term
=
= 0+1
=1
When n=4
Fifth term
=
= 1+2
=3
Therefore, the first five terms of the sequence defined by each of these recurrence relations and initial conditions is 1,2,0,1,3.
