Question

The same sequence may be defined in three ways: 1) f(x) = 5x + 1 for x = {1, 2, 3, ...}, 2) a1 = 6 and an + 1 = (an)(5) for n = {1, 2, 3, ...} and 3) an = 6 + 5(n − 1) for n = {1, 2, 3, ...}. They are all supposed to generate the same values, but one version may contain an error. If one version contains an error, identify the one with the error and fix it. A) None of these have errors, they all generate the same sequence. B) f(x) = 5x + 1 for x = {1, 2, 3, ...} It should say f(x) = 1x + 5 for x = {1, 2, 3, ...}. C) an = 6 + 5(n − 1) for n = {1, 2, 3, ...} It should say an = 5 + 6(n − 1) for n = {1, 2, 3, ...}. D) a1 = 6 and an + 1 = (an)(5) for n = {1, 2, 3, ...} It should say a1 = 6 and an + 1 = an + 5 for n = {1, 2, 3, ...}.

Answer 1

The question is:

The same sequence may be defined in three ways:

(1) f(x) = 5x + 1

for x = {1, 2, 3, ...}

(2) a_1 = 6 and a_(n + 1) = (a_n)(5)

for n = {1, 2, 3, ...}

(3) a_n = 6 + 5(n − 1)

for n = {1, 2, 3, ...}.

They are all supposed to generate the same values, but one version may contain an error. If one version contains an error, identify the one with the error and fix it.

(A) None of these have errors, they all generate the same sequence.

(B) f(x) = 5x + 1

for x = {1, 2, 3, ...}

It should say

f(x) = 1x + 5 for x = {1, 2, 3, ...}.

(C) a_n = 6 + 5(n − 1)

for n = {1, 2, 3, ...}

It should say a_n = 5 + 6(n − 1)

for n = {1, 2, 3, ...}.

(D) a_1 = 6 and a_(n + 1) = (a_n)(5) for n = {1, 2, 3, ...}

It should say a_1 = 6 and a_(n + 1) = a_n + 5

for n = {1, 2, 3, ...}.

Answer:

The correct option is (D)

Number (2) should say a_1 = 6 and a_(n + 1) = a_n + 5

for n = {1, 2, 3, ...}.

Step-by-step explanation:

Since the sequence may be defind in three ways, let us see the sequences generated by each of the representations, to see which one differs, and then correct it.

(1) f(x) = 5x + 1

for x = {1, 2, 3, ...}

Putting x = 1, 2, 3, 4, ...

f(1) = 5(1) + 1 = 6

f(2) = 5(2) + 1 = 11

f(3) = 5(3) + 1 = 16

f(4) = 5(4) + 1 = 21

And so on.

So, this generates a sequence:

6, 11, 16, 21, ...

(2) a_1 = 6 and a_(n + 1) = (a_n)(5)

for n = {1, 2, 3, ...}

Putting n = 1, 2, 3, 4, ...

n = 1

a_2 = (a_1)(5)

Since a_1 = 6

a_2 = 6 × 5 = 30

n = 2

a_3 = (a_2)(5)

= 30 × 5 = 150

n = 3

a_4 = (a_3)(5)

= 150 × 5 = 750

n = 4

a_5 = (a_4)(5)

= 750 × 5 = 3750

and so on.

The sequence generated is:

6, 30, 150, 3750, ...

(3) a_n = 6 + 5(n − 1)

for n = {1, 2, 3, ...}

Putting n = 1, 2, 3, 4, ...

n = 1

a_1 = 6 + 5(0) = 6

n = 2

a_2 = 6 + 5(1) = 11

n = 3

a_3 = 6 + 5(2) = 16

n = 4

a_4 = 6 + 5(3) = 21

and so on.

The sequence generated is:

6, 11, 16, 21, ...

Comparing the sequences generated by each of the notations, the sequences generated by

a_1 = 6 and a_(n + 1) = (a_n)(5)

for n = {1, 2, 3, ...}

differs from the rest, so it is the one with a problem.

It should be

a_1 = 6 and a_(n + 1) = a_n + 5

for n = {1, 2, 3, ...}.

Answer 2

Answer:

D) $a_{1}=6$ and $a_{n+1}=(a_{n})(5)$ for n = {1,2,3...}

It should say $a_{1}=6$ and $a_{n+1}=a_{n} +5$ for n = {1,2,3...}

Step-by-step explanation: