Stroud-Programme-10-Solutions.pdf

Engineering Mathematics, 7 th edition, edition, ISBN 9781137031204 Programme 10: Sequences

Test exercise 10 1 (a) Find the next two terms terms and and form form of the general general term term for the following following sequence: 1, 3, 5, 7, . . .

Solution: Solution: Step 1: 1: The numbers increase by 2 at a time so the next two numbers in the sequence are 7 + 2 = 9 and 9 + 2 = 11 Step 2: 2: The form of the general term is obtained by recognizing that this is an arithmetic sequence: a + nd where the first term is a = 1 and the common difference is d = = 2. Step 3: 3: The general form is then: 1 + 2n 2n

Engineering Mathematics seventh Mathematics seventh edition © K.A. Stroud and Dexter J. Booth 2013, published by Palgrave Macmillan, ISBN: 9781137031204

1 (b) Find the next two terms and form form of the general term for for the following sequence:

Solution: Solution: Step 1: 1: The numbers increase by 3 at a time so the next two numbers in the sequence are 5 + 3 = 8 and 8 + 3 = 11 Step 2: The form of the general term is obtained by recognizing that this is an arithmetic sequence: a + nd where the first term is

and the common difference is d = = 3.

Step 3: The general form is then:


1 (b) Find the next two terms and form form of the general term for for the following sequence:

Solution: Solution: Step 1: 1: The numbers increase by 3 at a time so the next two numbers in the sequence are 5 + 3 = 8 and 8 + 3 = 11 Step 2: The form of the general term is obtained by recognizing that this is an arithmetic sequence: a + nd where the first term is

and the common difference is d = = 3.



1 (c) Find the next next two terms and form of the general general term term for the following sequence: 16, 8, 4, 2, . . . Solution: Solution: Step 1: The numbers decrease by a half at a time so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a geometric sequence:

where the first term is

and the common ratio is

.



1 (d) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: Each number in the sequence is obtained by multiplying the previous number by minus 2 so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a geometric sequence:



.


Engineering Mathematics seventh edition © K.A. Stroud and Dexter J. Booth 2013, published by Palgrave Macmillan, ISBN: 9781137031204

1 (e) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: By inspection it is seen that each number in the sequence is obtained by adding the three previous numbers so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a recursive sequence of order three and so needs three starting values: where


1 (f) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: By inspection it is seen that each number in the sequence is obtained by adding the two previous numbers and multiplying that sum by 2 so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a recursive sequence of order two and so needs two starting values: where


3 (a) Find the recursive description corresponding to the following prescription for the output of a sequence:

Solution: Step 1:

Step 2: Therefore the recursive form is: where

(found from the prescription)


3 (b) Find the recursive description corresponding to the following prescription for the output of a sequence:

Solution: Step 1:




3 (c) Find the recursive description corresponding to the following prescription for the output of a sequence:

Solution: Step 1:




5

(a) Evaluate

Solution: Step 1:

Step 2:

Step 3:


5

(b) Evaluate

Solution: Step 1: because 5 > 1


5

(c) Evaluate

Solution: Step 1:

Step 2:

because 0 < 0.999 < 1


5

(d) Evaluate

Solution: Step 1:

Step 2:

Because the sequence just oscillates between


5

(e) Evaluate

Solution: Step 1:

Step 2:

Step 3:

Step 4: So


5

(f) Evaluate

Solution: Step 1:

Step 2:

Step 3: So


Further problems 10 1 (a) Find the next two terms and form of the general term for the following sequence: 1, 6, 11, 16, . . . Solution: Step 1: The numbers increase by 5 at a time so the next two numbers in the sequence are 16 + 5 = 21 and 21 + 5 = 26 Step 2: The form of the general term is obtained by recognizing that this is an arithmetic sequence: a + nd where the first term is a = 1 and the common difference is d = 5. Step 3: The general form is then: 1 + 5n


1 (b) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: The numbers increase by 2 at a time so the next two numbers in the sequence are

Step 2: The form of the general term is obtained by recognizing that this is an arithmetic sequence: a + nd where the first term is

and the common difference is d = 2.



1 (c) Find the next two terms and form of the general term for the following sequence: 10, 1, 0.1, 0.01, . . . Solution: Step 1: Each term is one tenth the previous term so the next two numbers in the sequence are

Step 2: The form of the general term is obtained by recognizing that this is a geometric sequence:

where the first term is 0


.



1 (d) Each term is one tenth the previous term so the next two numbers in the sequence are 1234.5, 123.45, 12.345, . . . Solution: Step 1: Each number in the sequence is obtained by dividing the previous number by 10 so the next two numbers in the sequence are 1.2345 and 0.12345 Step 2: The form of the general term is obtained by recognizing that this is a geometric sequence:



.



1 (e) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: By inspection it is seen that each number in the sequence is obtained by subtracting the previous term from the one before that so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a recursive sequence of order two and so needs two starting values: where


1 (f) Find the next two terms and form of the general term for the following sequence:

Solution: Step 1: By inspection it is seen that each number in the sequence is obtained by adding the two previous numbers so the next two numbers in the sequence are and Step 2: The form of the general term is obtained by recognizing that this is a recursive sequence of order two and so needs two starting values: where


3

(a) Solve the difference equation:

Solution: Step 1: Assuming

So that

then substituting into the difference equation gives:

.

Step 2: The general form of the solution to the difference equation is then:

Step 3: Applying the given conditions

: [1] [2]

From [1]

and substituting into [2] gives

.

Step 4: Therefore


3

(b) Solve the difference equation:



So that

.



: [1] [2]

From

[1]

and

substituting

into

[2]

gives

and therefore Step 4: Therefore


3

(c) Solve the difference equation:



So that

.



: [1] [2]

From

[1]

and

substituting

into

[2]

gives

and therefore Step 4: Therefore


5

(a) Does the sequence

converge or diverge?

Solution: Step 1:

Step 2: Sequence diverges


5

(b) Does the sequence


Solution: Step 1:

Step 2: Sequence diverges


5

(c) Does the sequence


Solution: Step 1: Assume that the sequence converges to A. Then:

That is: so that Step 2: Sequence converges to


5

(d) Does the sequence


Solution: Step 1: Since

then

and so sequence converges

Step 2: Sequence converges to 0


5

(e) Does the sequence


Solution: Step 1:

Each term of the sequence is 10 times greater than the previous term. Step 2: Sequence diverges


7 The first, third and sixth terms of an arithmetic sequence form three successive terms of a geometric sequence. If the first term of both the arithmetic and geometric sequence is 8, find the second, third and fourth terms and general term of the geometric sequences. Solution: Step 1: The terms 8, 8 + 2d , 8 + 5d are the first, third and sixth terms of the arithmetic sequence in question and are three successive terms of the corresponding geometric sequence. Step 2: So:

Step 3: That is: and so Step 4: Therefore: and so The common ratio is then

.

Step 5: The second, third and fourth terms of the geometric sequence are then:


9

For what value of

does

differ from 2 by 0.02 to 2 dp?

Solution: Step 1: When n = 0 the sequence term has the value

and:

Step 2: So the terms of the sequence are increasing but are always less than 2. So we need to know the value of n corresponding to a term value of 1.975, this being the first value that is equal to 1.98 to 2 dp: Step 3: That is

that is

Step 4: This gives: so that


11 If three numbers in geometric sequence are respectively subtracted from three numbers also in geometric sequence leaving remainders that are also in geometric sequence show that all three geometric sequences have the same common ratio. Solution: Step 1: Let three sequential numbers from two geometric sequences be given as: and Step 2: Then it is given that geometric sequence. That is if

also form three terms of another

then

and

.

Step 3: Furthermore: and Step 4: Therefore that is Step 5: So that:

Step 6: That is: so Step 7: Engineering Mathematics seventh edition © K.A. Stroud and Dexter J. Booth 2013, published by Palgrave Macmillan, ISBN: 9781137031204

Therefore: and so That is,

so all three common ratios are the same.


13 If x , 5 x , 6 x + 9 form three successive terms of an arithmetic sequence, find the next four terms. Solution: Step 1: Let Step 2: Therefore:

Step 3: That is:

Step 4: The three given terms of the arithmetic sequence are then 3, 15, 27 and the next four are: 39, 51, 63, 75


15

The twelfth and nineteenth terms of an harmonic sequence are respectively and

. Find the fourth term.

Solution: Step 1: The reciprocals of an harmonic sequence form an arithmetic sequence so the twelfth and nineteenth terms of the corresponding arithmetic sequence are: 5 and

respectively.

Step 2: That is:

Step 3: Therefore: and so

and

Step 4: The fourth term of the arithmetic sequence is then

Step 5: The fourth term of the harmonic sequence is then


17

If

form three successive terms of an arithmetic sequence

show that sequence.

also form three successive terms of another arithmetic

Solution: Step 1: Let:

Then:

Step 2: That is:

Step 3: So that:

Step 4: And hence: that is Step 5: Therefore if: then

Step 6: Engineering Mathematics seventh edition © K.A. Stroud and Dexter J. Booth 2013, published by Palgrave Macmillan, ISBN: 9781137031204

That is: form three successive terms of an arithmetic sequence


19

Show that the difference equation

can be derived

from the coupled difference equations:

Find

and

given that

and

.

Solution: Step 1: and so Step 2: As a consequence:

Step 3: Since then

, that is

Step 4: Assume

and substitute into the difference equation to obtain:

That is:

Step 5: Therefore: and so Step 6: Applying the conditions: Engineering Mathematics seventh edition © K.A. Stroud and Dexter J. Booth 2013, published by Palgrave Macmillan, ISBN: 9781137031204

Stroud-Programme-10-Solutions.pdf

Recommend Documents