Secrets

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Question 1

A. This is an arithmetic pattern.

B. This is a geometric pattern.

C. When graphed on the coordinate plane, this pattern would model a linear function.

D. When graphed on the coordinate plane, this pattern would model an exponential function.

E. The formula a_n = 4 + (n - 1)(3)a sub n equals 4 plus the quantity n minus 1 times three models this pattern.

F. The formula a_n = 4times⋅3^{n - 1}a sub n equals 4 times 3 to the power of n minus 1 models this pattern.

G. The equation y = 3x + 1y equals 3x plus 1 models this pattern.

H. The equation y = 4times⋅3^{x - 1}y equals 4 times 3 to the power of x minus 1 models this pattern.

I. The graph below models this pattern.

J. The graph below models this pattern.

Answer: Correct! Go to question 2.

Answer: Incorrect. The answers are: A, C, E, G and I.

The pattern is an arithmetic sequence with a common difference of +3.

Using the table entries as ordered pairs, the resulting graph would be linear.

Following the formula for an arithmetic sequence:
a_n = a₁ + (n - 1)da sub n equals a sub 1 plus the quantity n minus 1 times d.
a_n = 4 + (n - 1)times⋅3a sub n equals 4 plus the quantity n minus 1 times 3

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Arithmetic sequences are linear relationships, where the number of the term is the input, x, and the value of the terms is the output, y.

y = 4 + (x - 1)times⋅3y equals 4 plus the quantity x minus 1 times 3
y = 4 + 3x - 3y equals 4 plus 3x minus 3
y = 3x + 1y equals 3x plus 1

Go to question 2.

Answer: One or more of your answers are incorrect. Review the Learn It section, and then try again.

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Question 2

A. This is an arithmetic pattern.

B. This is a geometric pattern.

C. When graphed on the coordinate plane, this pattern would model a linear function.

D. When graphed on the coordinate plane, this pattern would model an exponential function.

E. The formula a_n = 2 + (n - 1)(2)a sub n equals 2 plus the quantity n minus 1 times 2 models this pattern.

F. The formula a_n = 2times⋅2^{n - 1}a sub n equals 2 times 2 to the power of n minus 1 models this pattern.

G. The equation y = 2x + 2y equals 2x plus 2 models this pattern.

H. The equation y = 2^xy equals 2 to the power of x models this pattern.

I. The graph below models this pattern.

J. The graph below models this pattern.

Answer: Correct! Go to question 2.

Answer: Incorrect. The answers are: B, D, F, H and J.

The pattern is a geometric sequence with a common ratio of 2.

Using the table entries as ordered pairs, the resulting graph would be exponential.

Following the formula for a geometric sequence:
a_n = a₁times⋅r^n-1
a_n = 2times⋅2^n-1
y = 2times⋅2^x-1
y = 2times⋅2^xtimes⋅2^-1
y = 2times⋅2^xtimes⋅
y = 2^x

a sub n equals a sub 1 times r to the power of n minus 1. a sub n equals 2 times 2 to the power of n minus 1. y equals 2 times 2 to the power of x minus 1. y equals 2 times 2 to the power of x times 2 to the power of negative 1. y equals 2 times 2 to the power of x times one half. y equals 2 to the power of x. Go to question 3.

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Question 3

Nathan posts a secret on Facebook, and sixty of his friends read it. However, each day after the post, of the friends will forget the secret. The number of people who remember the secret follows a geometric sequence, and can therefore be modeled using an exponential function.

Which of the following geometric formulas and corresponding exponential functions models this situation?

A. a_n = 60times⋅^n-1 and y = 360times⋅^x a sub n equals 60 times one sixth to the power of n minus one and y equals 360 times one sixth to the power of x B. a_n = times⋅60^n-1 and y = times⋅360^x a sub n equals one sixth times sixty to the power of n minus one and y equals one sixth times 360 to the power of x C. a_n = 6times⋅^n-1 and y = 60times⋅^x a sub n equals 6 times one sixth to the power of n minus one and y equals 60 times one sixth to the power of x D. a_n = 360times⋅^n-1 and y = 6times⋅^x a sub n equals 360 times one sixth to the power of n minus one and y equals 6 times one sixth to the power of x

Answer: Correct! Go to question 4.

Answer: Incorrect. The answer is A.

a_n = 60times⋅^n-1
a_n = 60times⋅()ⁿ()^-1
a_n = 60times⋅()ⁿ(6)
a_n = 360times⋅()ⁿ a sub n equals 60 times one sixth to the power of n minus 1. a sub n equals 60 times one sixth to the power of n times one sixth to the power of negative 1. a sub n equals 60 times one sixth to the power of n times 6. a sub n equals 360 times one sixth to the power of n.
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y = 360times⋅()^x y equals 360 times one sixth to the power of x

(This is exponential decay because the absolute value of the common ratio is less than one.)

Go to question 4.

Answer:Incorrect. Review the Learn It section, and then try again.

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Question 4

The two arithmetic representations model the growth of a rumor spread around a school. Which sequence has the greater 6^th term?

A. a_n=53+(n-1)(-negative7) a sub n equals 53 plus the quantity of n minus 1 times negative 7 B.

Answer: Correct!

Answer: Incorrect. The answer is B.
The sixth term for a_n=53+(n-1)(-negative7)a sub n equals 53 plus the quantity of n minus 1 times negative 7 would be a₆=53+(6-1)(-negative7)=18a sub 6 equals 53 plus the quantity of 6 minus 1 times negative 7 equals 18.
Extending the graph for answer choice B shows the sixth term to be 23.

Answer:Incorrect. Review the Learn It section, and then try again.