Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions.

a)

b)

c)

d)

e)

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.