Question 1

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 2

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Show that the formula2 + 4 + 6 + 8 + ... + 2n =  + n + 3obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for sonatural number k, it is also true for the next natural number k + 1. Then show that the formula is false forn = 1.

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 3

Solve.
-A ping-pong ball is dropped from a height of 9 ft and always rebounds  of the distance fallen. Find thetotal sum of the rebound heights of the ball.
A)4.5 ft
B)13.5 ft
C)6 ft
D)3 ft

Accepted Answer

The answer of Solve.
-A ping-pong ball is dropped from a...

Question 4

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 5

Solve.
-Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays6.5% per year compounded annually. You plan to invest $2,500 at the end of each year for the next 10years. How much will your account have in it at the end of 10 years?
A)$33,736
B)$35,506
C)$32,193
D)$35,034

Accepted Answer

The answer of Solve.
-Looking ahead to retirement, you sign up...

Question 6

Solve.
-After working for 25 years you would like to have $800,000 in an annuity for early retirement. If theannual interest rate is 7.5%, compounded monthly, what will your monthly deposit need to be?
A)$911.93
B)$2666.67
C)$3088.48
D)$1146.98

Accepted Answer

To find the monthly deposit required to reach a future value of $800,000 in an annuity, we use the future value of an annuity formula: $FV = P \times \frac{(1 + r)^n - 1}{r}$, where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the total number of payments. Rearranging the formula to solve for P gives us $P = FV \times \frac{r}{(1 + r)^n - 1}$. Given an annual interest rate of 7.5%, the monthly interest rate is $0.075/12 = 0.00625$. The total number of payments for 25 years with monthly deposits is $25 \times 12 = 300$. Plugging these values into the formula gives $P = 800,000 \times \frac{0.00625}{(1 + 0.00625)^{300} - 1}$, which calculates to approximately $911.93.

Question 7

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 8

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Use the Principle of Mathematical Induction to show that the statement ʺ5 is a factor of  ʺ is true forall natural numbers. (

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 9

Solve.
-To save for retirement, you decide to deposit $2,500 into an IRA at the end of each year for the next 30years. If the interest rate is 7% per year compounded annually, find the value of the IRA after 30 years.(Round to the nearest dollar.)
A)$236,152
B)$16,531
C)$218,366
D)$4,691,339

Accepted Answer

The value of the IRA after 30 years can be calculated using the future value of an annuity formula: FV = P * [((1 + r)^n - 1) / r], where P is the annual payment ($2,500), r is the annual interest rate (7% or 0.07), and n is the number of payments (30 years). Plugging in the values, we get FV = $2,500 * [((1 + 0.07)^30 - 1) / 0.07], which rounds to approximately $236,152.

Question 10

Solve.
-A pendulum bob swings through an arc 70 inches long on its first swing. Each swing thereafter, it swingsonly 70% as far as on the previous swing. How far will it swing altogether before coming to a completestop?
A)233 inches
B)100 inches
C)117 inches
D)200 inches

Accepted Answer

The total distance the pendulum swings can be found using the formula for the sum of a geometric series, S = a / (1 - r), where a is the first term (70 inches) and r is the common ratio (0.7). Plugging these values in gives S = 70 / (1 - 0.7) = 70 / 0.3 = 233.33 inches. Since the question likely assumes a rounding to the nearest whole number, the answer is 233 inches.

Question 11

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 12

Solve.
-Domenica invests $250 each quarter in a fixed-interest mutual fund paying annual interest of 7%compounded quarterly. How much will her account have in it at the end of 11 years?
A)$16,363
B)$3,946
C)$49,343
D)$16,492

Accepted Answer

The investment scenario described is a case of a future value of a series in compound interest. The formula to calculate the future value of a series is $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the payment per period, $r$ is the interest rate per period, and $n$ is the total number of payments. Given an annual interest rate of 7% compounded quarterly, the quarterly interest rate is $7\% / 4 = 1.75\%$, or $0.0175$ in decimal form. Domenica invests $250 each quarter for 11 years, which equals $11 \times 4 = 44$ quarters. Plugging these values into the formula gives $FV = 250 \times \frac{(1 + 0.0175)^{44} - 1}{0.0175}$, which calculates to approximately $16,363.

Question 13

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Show that the statement ʺ  is a prime numberʺ is true for n = 1, but is not true for n = 3.

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 14

Solve.
-Lonnie deposits $150 each month into an account paying annual interest of 7% compounded monthly.How much will his account have in it at the end of 14 years?
A)$42,606
B)$3,383
C)$42,452
D)$42,735

Accepted Answer

The account balance can be calculated using the formula for the future value of a series: $FV = P \times \frac{(1 + r)^n - 1}{r}$, where $P$ is the monthly deposit, $r$ is the monthly interest rate (annual rate divided by 12), and $n$ is the total number of deposits (months). Here, $P = $150$, $r = 0.07/12$, and $n = 14 \times 12$. Plugging these values into the formula gives the future value, which matches choice A, $42,606.

Question 15

Solve.
-Jack decided to put $600 into an IRA account every 3 months at a rate of 6% compounded quarterly. Find arecursive formula that represents his balance at the end of each quarter. How long will it be before thevalue of the account is $100,000? What will be the balance in 30 years when Jack retires?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Solve.
-Jack decided to put $600 into an...

Question 16

Solve.
-A ball is dropped from a height of 25 feet. Each time it strikes the ground, it bounces up to 0.7 of theprevious height. The total distance the ball has traveled before the second bounce is 25 + 2(25 ∙ 0.7)feet,and the total distance the ball has traveled before bounce n + 1 is  Use facts about infinite geometric series to calculate the total distance the ball has traveled by the time it hasstopped bouncing.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Solve.
-A ball is dropped from a height...

Question 17

Mathematical Induction
1 Prove Statements Using Mathematical Induction
 Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-

Accepted Answer

The answer of  Mathematical Induction
1 Prove Statements Using Mathematical...

Question 18

Write the word or phrase that best completes each statement or answers the question.
-Christine contributes $100 each month to her 401(k). To the nearest dollar, what will be the value ofChristineʹs 401(k)in 20 years if the per annum rate of return is assumed to be 10% compounded monthly.

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Solve.
-After being struck with a hammer, a gong vibrates 45 vibrations in the first second and in each secondthereafter makes  as many vibrations as in the previous second. Find how many vibrations the gongmakes before it stops vibrating.
A)270 vibrations
B)54 vibrations
C)280 vibrations
D)49 vibrations

Accepted Answer

The answer of Solve.
-After being struck with a hammer, a...

Question 20

Write the word or phrase that best completes each statement or answers the question.
-You deposit $150 each 6 months into an annuity with an annual interest rate of 13.5%, compoundedsemiannually.a)What is the balance after 20 years?b)What is the balance after 40 years?

Accepted Answer

The answer of Write the word or phrase that best...

Quiz 9: Sequences; Induction; the Binomial Theorem

Mathematical Induction 1 Prove Statements Using Mathematical Induction Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. -

Solve. -A ping-pong ball is dropped from a height of 9 ft and always rebounds of the distance fallen. Find thetotal sum of the rebound heights of the ball.

Mathematical Induction 1 Prove Statements Using Mathematical Induction Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. -

Solve. -Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays6.5% per year compounded annually. You plan to invest $2,500 at the end of each year for the next 10years. How much will your account have in it at the end of 10 years?

Solve. -After working for 25 years you would like to have $800,000 in an annuity for early retirement. If theannual interest rate is 7.5%, compounded monthly, what will your monthly deposit need to be?

Mathematical Induction 1 Prove Statements Using Mathematical Induction Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. -

Solve. -To save for retirement, you decide to deposit $2,500 into an IRA at the end of each year for the next 30years. If the interest rate is 7% per year compounded annually, find the value of the IRA after 30 years.(Round to the nearest dollar.)

Solve. -A pendulum bob swings through an arc 70 inches long on its first swing. Each swing thereafter, it swingsonly 70% as far as on the previous swing. How far will it swing altogether before coming to a completestop?

Mathematical Induction 1 Prove Statements Using Mathematical Induction Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. -

Solve. -Domenica invests $250 each quarter in a fixed-interest mutual fund paying annual interest of 7%compounded quarterly. How much will her account have in it at the end of 11 years?

Solve. -Lonnie deposits $150 each month into an account paying annual interest of 7% compounded monthly.How much will his account have in it at the end of 14 years?

Mathematical Induction 1 Prove Statements Using Mathematical Induction Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. -

Write the word or phrase that best completes each statement or answers the question. -Christine contributes $100 each month to her 401(k). To the nearest dollar, what will be the value ofChristineʹs 401(k)in 20 years if the per annum rate of return is assumed to be 10% compounded monthly.

Solve. -After being struck with a hammer, a gong vibrates 45 vibrations in the first second and in each secondthereafter makes as many vibrations as in the previous second. Find how many vibrations the gongmakes before it stops vibrating.

Write the word or phrase that best completes each statement or answers the question. -You deposit $150 each 6 months into an annuity with an annual interest rate of 13.5%, compoundedsemiannually.a)What is the balance after 20 years?b)What is the balance after 40 years?