Find the first five terms of the sequence defined by each

Question

Problem 9E

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

a) ${a}_{n}\ =\ 6{a}_{n-1},\ {a}_{0}\ =\ 2$

b) ${a}_{n}\ =\ {a}_{n-1}^{2},\ {a}_{1}\ =\ 2$

c) ${a}_{n}\ =\ {a}_{n-1}^{\ }\ +\ 3{a}_{n-2},\ {a}_{0}\ =\ 1,\ {a}_{1}\ =\ 2$

d) ${a}_{n}\ =\ {{na}_{n-1}^{\ }\ +\ n}_{\ }^{2}{a}_{n-2},\ {a}_{0}\ =\ 1,{\ a}_{1}\ =\ 1$

e) ${a}_{n}\ =\ {a}_{n-1}^{\ }\ +\ {a}_{n-3},\ {a}_{0}\ =\ 1,\ {a}_{1}\ =\ 2,\ {a}_{2}\ =\ 0$

Answer 1

Step-by-Step Solution:

Step 1 of 3

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. ${a}_{n}^{\ }=6{a}_{n-1}^{\ },\ {a}_{0}^{\ }=2$

First term ${a}_{0}^{\ }=2$

When n=1

Second term

${a}_{1}^{\ }=6{a}_{1-1}^{\ }$

= $6{a}_{0}^{\ }$

= 6.2

= 12

When n=2

Third term

${a}_{2}^{\ }=6{a}_{2-1}^{\ }$

= $6{a}_{1}^{\ }$

= 6.12

= 72

When n=3

Fourth term

${a}_{3}^{\ }=6{a}_{3-1}^{\ }$

= $6{a}_{2}^{\ }$

= 6.72

= 432

When n=4

Fifth term

${a}_{4}^{\ }=6{a}_{4-1}^{\ }$

= $6{a}_{3}^{\ }$

= 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. ${a}_{n}^{\ }={a}_{n-1}^{2},\ {a}_{1}^{\ }=2$

First term ${a}_{1}^{\ }=2$

When n=2

Second term

${a}_{2}^{\ }={a}_{2-1}^{2}$

= ${a}_{1}^{2}$

= ${\left({2}\right)}_{\ }^{2}$

=4

When n=3

Third term

${a}_{3}^{\ }={a}_{3-1}^{2}$

= ${a}_{2}^{2}$

= ${\left({4}\right)}_{\ }^{2}$

=16

When n=4

Fourth term

${a}_{4}^{\ }={a}_{4-1}^{2}$

= ${a}_{3}^{2}$

= ${\left({16}\right)}_{\ }^{2}$

=256

When n=5

Fifth term

${a}_{5}^{\ }={a}_{5-1}^{2}$

= ${a}_{4}^{2}$

= ${\left({256}\right)}_{\ }^{2}$

=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. ${a}_{n}^{\ }={a}_{n-1}^{\ }+3{a}_{n-2}^{\ },\ {a}_{0}^{\ }=1,\ {a}_{1}^{\ }=2$

First term ${a}_{0}^{\ }=1$

Second term ${a}_{1}^{\ }=2$

When n=2

Third term

${a}_{2}^{\ }={a}_{2-1}^{\ }+3{a}_{2-2}^{\ }$

= ${a}_{1}^{\ }+3{a}_{0}^{\ }$

=2+3.1

=2+3

=5

When n=3

Fourth term

${a}_{3}^{\ }={a}_{3-1}^{\ }+3{a}_{3-2}^{\ }$

= ${a}_{2}^{\ }+3{a}_{1}^{\ }$

=5+3.2

=5+6

=11

When n=4

Fifth term

${a}_{4}^{\ }={a}_{4-1}^{\ }+3{a}_{4-2}^{\ }$

= ${a}_{3}^{\ }+3{a}_{2}^{\ }$

=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. ${a}_{n}^{\ }=n{a}_{n-1}^{\ }+{n}_{\ }^{2}{a}_{n-2}^{\ },\ {a}_{0}^{\ }=1,{a}_{1}^{\ }=1$

First term ${a}_{0}^{\ }=1$

Second term ${a}_{1}^{\ }=1$

When n=2

Third term

${a}_{2}^{\ }=2{a}_{2-1}^{\ }+{2}_{\ }^{2}{a}_{2-2}^{\ }$

= $2{a}_{1}^{\ }+{2}_{\ }^{2}{a}_{0}^{\ }$

=2.1+4.1

= 2+4

=6

When n=3

Fourth term

${a}_{3}^{\ }=3{a}_{3-1}^{\ }+{3}_{\ }^{2}{a}_{3-2}^{\ }$

= $3{a}_{2}^{\ }+{3}_{\ }^{2}{a}_{1}^{\ }$

=3.6+9.1

= 18+9

=27

When n=4

Fifth term

${a}_{4}^{\ }=4{a}_{4-1}^{\ }+{4}_{\ }^{2}{a}_{4-2}^{\ }$

= $4{a}_{3}^{\ }+{4}_{\ }^{2}{a}_{2}^{\ }$

=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. ${a}_{n}^{\ }={a}_{n-1}^{\ }+{a}_{n-3}^{\ },\ {a}_{0}^{\ }=1,{a}_{1}^{\ }=2,{a}_{2}^{\ }=0$

First term ${a}_{0}^{\ }=1$

Second term ${a}_{1}^{\ }=2$

Third term ${a}_{2}^{\ }=0$

When n=3

Fourth term

${a}_{3}^{\ }={a}_{3-1}^{\ }+{a}_{3-3}^{\ }$

= ${a}_{2}^{\ }+{a}_{0}^{\ }$

= 0+1

=1

When n=4

Fifth term

${a}_{4}^{\ }={a}_{4-1}^{\ }+{a}_{4-3}^{\ }$

= ${a}_{3}^{\ }+{a}_{1}^{\ }$

= 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.

Step 2 of 3

Chapter 2.4, Problem 9E is Solved

Step 3 of 3

Find the first five terms of the sequence defined by each

Solution for problem 9E Chapter 2.4

Chapter 2.4, Problem 9E is Solved

Textbook: Discrete Mathematics and Its Applications

Edition: 7

Author: Kenneth Rosen

ISBN: 9780073383095

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