Question 1

Prove the following statement by mathematical induction: $3+7+11+\cdots+(4 n-1)=n(2 n+1)$

Accepted Answer

$P_{1}=3=1(2 \cdot 1+1)$

Assume $ P_{k}: 3+7+11+\ldots+(4 k-1) $ $ =k(2 k+1) $

$\text { Prove } P_{k+1}=(k+1)(2(k+1)+1)$

Question 2

Determine the sixth partial sum of the geometric sequence $5, \frac{5}{4}, \frac{5}{16}, \ldots$
A) $\frac{5}{4096}$
B) $\frac{1}{1024}$
C) $\frac{5}{1024}$
D)  6.48 
E)  6.67

Accepted Answer

The sixth partial sum of a geometric sequence can be found using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. Here, $a = 5$, $r = \frac{1}{4}$, and $n = 6$. Plugging these values into the formula gives $S_6 = \frac{5(1 - (\frac{1}{4})^6)}{1 - \frac{1}{4}} = \frac{5(1 - \frac{1}{4096})}{\frac{3}{4}} = \frac{5 \cdot \frac{4095}{4096}}{\frac{3}{4}} = \frac{5 \cdot 4095}{3072} = \frac{20475}{3072} \approx 6.67$.

Question 3

Write the statements P₁, P_k , and P_k+1 for the equation $1(1 !)+2(2 !)+\cdots+n(n !)=(n+1) !-1$ as if you were writing a proof by mathematical induction. Do not write a proof.

Accepted Answer

$\begin{array}{l}
P_{1}=1(1 !)=2 !-1 \
P_{k}=1(1 !)+2(2 !)+\ldots+k(k !)=(k+1) !-1 \
P_{k+1} \
=1(1 !)+2(2 !)+\ldots+k(k !)+(k+1)(k+1) ! \
=(k+1) !-1+(k+1)(k+1) !
\end{array}$

Question 4

Find the coefficient of the $x^{3} y^{4}$ term in the expansion of $(x+y)^{7}$
A) 3
B) 4
C) $7 !$
D) 35
E) 210

Accepted Answer

The coefficient of $x^{3}y^{4}$ in the expansion of $(x+y)^{7}$ can be found using the binomial theorem, specifically the term is given by $\binom{7}{3}x^{3}y^{4}$. The binomial coefficient $\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.

Question 5

Find the sum of the first 12 terms of the arithmetic sequence:
 28,22,16,10, ....

Accepted Answer

The answer of Find the sum of the first 12...

Question 6

Find the sum of the infinite series $\sum_{k=1}^{\infty} 4\left(\frac{1}{5}\right)^{k-1}$

Accepted Answer

The answer of Find the sum of the infinite series...

Question 7

Assume that the probability that a newborn child is a female is  50% . In a family of four children, what is the probability that
(a) all the children are girls?
(b) at least two of the children are boys?

Accepted Answer

The answer of Assume that the probability that a newborn...

Question 8

The third and fifth terms of a geometric sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

Accepted Answer

The answer of The third and fifth terms of a...

Question 9

The sum of the squares of the first  n  positive odd integers is  $\frac{(2 n-1)(2 n)(2 n+1)}{6}$ 
Use this expression to find the sum $1^{2}+3^{2}+5^{2}+\cdots+75^{2}$
A) 5628
B) 73,150
C) 1425
D)  7,700,625 
E) 5776

Accepted Answer

The sum of the squares of the first n positive odd integers is given by the formula $\frac{(2 n-1)(2 n)(2 n+1)}{6}$. To find the sum $1^{2}+3^{2}+5^{2}+\cdots+75^{2}$, we need to substitute n = 38 into the formula.$\frac{(2 \times 38-1)(2 \times 38)(2 \times 38+1)}{6} = \frac{(75)(76)(77)}{6} = 73,150$

Question 10

A fair coin is tossed five times. Find the probability of tossing exactly two tails in those five tosses.

Accepted Answer

The answer of A fair coin is tossed five times....

Question 11

Sally deposits $ \$ 100 $ at the end of each month into an account that pays $ 6 \% $ annual interest compounded monthly. After 5 years of deposits, what is the account balance?

Accepted Answer

The answer of Sally deposits $ \$ 100 $ at...

Question 12

Do the following sequences converge or diverge? If they converge, state the limit.
(a) $\left\{\frac{5 n}{2 n+1}\right\}$
(b) $\left\{\frac{n^{3}+1}{2 n+1}\right\}$
(c) $\left\{\frac{3 n^{3}}{4 n^{5}+1}\right\}$

Accepted Answer

The answer of Do the following sequences converge or diverge?...

Question 13

Suppose there is a  40%  chance of rain tomorrow. If it rains, there is a  20%  chance that all of the rides at an amusement park will be operating. If it doesn't rain, there is a  90%  chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

Accepted Answer

The answer of Suppose there is a  40% ...

Question 14

The third and fifth terms of an arithmetic sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

Accepted Answer

The answer of The third and fifth terms of an...

Question 15

A computer randomly selects a number from 1 through 50 . Two selections are made by the computer to create an ordered pair of numbers. How many elements are in the sample space for this experiment?
A) 2,500
B) 2,450
C) 2,000
D) 100
E) 50

Accepted Answer

Since the computer can select any number from 1 through 50 for each of the two selections, and the order matters (making it an ordered pair), there are 50 choices for the first number and 50 choices for the second number. Multiplying these together gives 50 * 50 = 2,500 possible ordered pairs.

Question 16

A computer password is 6 entries long. If the password must have 3 letters followed by 3 numbers with repetition of numbers and/or letters permitted, how many different passwords are possible?

Accepted Answer

The answer of A computer password is 6 entries long....

Question 17

A computer randomly selects a number from 1 through 50 . Two selections are made by the computer to create an ordered pair of numbers. 2500 elements are in the sample space for this experiment
-For the experiment above , what is the probability that the computer will not select the same number in both selections?

Accepted Answer

The answer of A computer randomly selects a number from...

Question 18

Find the fifteenth term of $(x+2 y)^{20}$ where $x^{20}$ is labelled the zeroth term.

Accepted Answer

The answer of Find the fifteenth term of $(x+2 y)^{20}$...

Question 19

Expand $(5 x-8 y)^{5}$

Accepted Answer

The answer of Expand $(5 x-8 y)^{5}$...

Question 20

There are 85 students in a college mathematics class. Five students are chosen to work on a project together. How many different groups of five students are there in this class?

Accepted Answer

The answer of There are 85 students in a college...

Quiz 9: Discrete Mathematics

Prove the following statement by mathematical induction: $3+7+11+\cdots+(4 n-1)=n(2 n+1)$

Determine the sixth partial sum of the geometric sequence $5, \frac{5}{4}, \frac{5}{16}, \ldots$

Write the statements P₁, P_k , and P_k+1 for the equation $1(1 !)+2(2 !)+\cdots+n(n !)=(n+1) !-1$ as if you were writing a proof by mathematical induction. Do not write a proof.

Find the coefficient of the $x^{3} y^{4}$ term in the expansion of $(x+y) ^{7}$

Find the sum of the first 12 terms of the arithmetic sequence: 28,22,16,10, ....

Find the sum of the infinite series $\sum_{k=1}^{\infty} 4\left(\frac{1}{5}\right)^{k-1}$

Assume that the probability that a newborn child is a female is 50% . In a family of four children, what is the probability that (a) all the children are girls? (b) at least two of the children are boys?

The third and fifth terms of a geometric sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

The sum of the squares of the first n positive odd integers is $\frac{(2 n-1) (2 n) (2 n+1) }{6}$ Use this expression to find the sum $1^{2}+3^{2}+5^{2}+\cdots+75^{2}$

A fair coin is tossed five times. Find the probability of tossing exactly two tails in those five tosses.

Sally deposits $\$ 100$ at the end of each month into an account that pays $6 \%$ annual interest compounded monthly. After 5 years of deposits, what is the account balance?

Do the following sequences converge or diverge? If they converge, state the limit. (a) $\left\{\frac{5 n}{2 n+1}\right\}$ (b) $\left\{\frac{n^{3}+1}{2 n+1}\right\}$ (c) $\left\{\frac{3 n^{3}}{4 n^{5}+1}\right\}$

Suppose there is a 40% chance of rain tomorrow. If it rains, there is a 20% chance that all of the rides at an amusement park will be operating. If it doesn't rain, there is a 90% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

The third and fifth terms of an arithmetic sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

A computer randomly selects a number from 1 through 50 . Two selections are made by the computer to create an ordered pair of numbers. How many elements are in the sample space for this experiment?

A computer password is 6 entries long. If the password must have 3 letters followed by 3 numbers with repetition of numbers and/or letters permitted, how many different passwords are possible?

Find the fifteenth term of $(x+2 y)^{20}$ where $x^{20}$ is labelled the zeroth term.

Expand $(5 x-8 y)^{5}$

There are 85 students in a college mathematics class. Five students are chosen to work on a project together. How many different groups of five students are there in this class?

Quiz 9: Discrete Mathematics

Prove the following statement by mathematical induction: 3+7+11+⋯+(4n−1)=n(2n+1)3+7+11+\cdots+(4 n-1)=n(2 n+1)3+7+11+⋯+(4n−1)=n(2n+1)

Determine the sixth partial sum of the geometric sequence 5,54,516,…5, \frac{5}{4}, \frac{5}{16}, \ldots5,45​,165​,…

Write the statements P1, Pk , and Pk+1 for the equation 1(1!)+2(2!)+⋯+n(n!)=(n+1)!−11(1 !)+2(2 !)+\cdots+n(n !)=(n+1) !-11(1!)+2(2!)+⋯+n(n!)=(n+1)!−1 as if you were writing a proof by mathematical induction. Do not write a proof.

Find the coefficient of the x3y4x^{3} y^{4}x3y4 term in the expansion of (x+y) 7(x+y) ^{7}(x+y) 7

Find the sum of the first 12 terms of the arithmetic sequence: 28,22,16,10, ....

Find the sum of the infinite series ∑k=1∞4(15)k−1\sum_{k=1}^{\infty} 4\left(\frac{1}{5}\right)^{k-1}∑k=1∞​4(51​)k−1

Assume that the probability that a newborn child is a female is 50% . In a family of four children, what is the probability that (a) all the children are girls? (b) at least two of the children are boys?

The third and fifth terms of a geometric sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

The sum of the squares of the first n positive odd integers is (2n−1) (2n) (2n+1) 6\frac{(2 n-1) (2 n) (2 n+1) }{6}6(2n−1) (2n) (2n+1) ​ Use this expression to find the sum 12+32+52+⋯+7521^{2}+3^{2}+5^{2}+\cdots+75^{2}12+32+52+⋯+752

A fair coin is tossed five times. Find the probability of tossing exactly two tails in those five tosses.

Sally deposits $100 \$ 100 $100 at the end of each month into an account that pays 6% 6 \% 6% annual interest compounded monthly. After 5 years of deposits, what is the account balance?

Do the following sequences converge or diverge? If they converge, state the limit. (a) {5n2n+1}\left\{\frac{5 n}{2 n+1}\right\}{2n+15n​} (b) {n3+12n+1}\left\{\frac{n^{3}+1}{2 n+1}\right\}{2n+1n3+1​} (c) {3n34n5+1}\left\{\frac{3 n^{3}}{4 n^{5}+1}\right\}{4n5+13n3​}

Suppose there is a 40% chance of rain tomorrow. If it rains, there is a 20% chance that all of the rides at an amusement park will be operating. If it doesn't rain, there is a 90% chance all of the rides will be operating. What is the probability that all of the rides will be operating tomorrow?

The third and fifth terms of an arithmetic sequence are 2 and 32, respectively. Find explicit and recursive formulas for the sequence.

A computer randomly selects a number from 1 through 50 . Two selections are made by the computer to create an ordered pair of numbers. How many elements are in the sample space for this experiment?

A computer password is 6 entries long. If the password must have 3 letters followed by 3 numbers with repetition of numbers and/or letters permitted, how many different passwords are possible?

Find the fifteenth term of (x+2y)20(x+2 y)^{20}(x+2y)20 where x20x^{20}x20 is labelled the zeroth term.

Expand (5x−8y)5(5 x-8 y)^{5}(5x−8y)5

There are 85 students in a college mathematics class. Five students are chosen to work on a project together. How many different groups of five students are there in this class?

Prove the following statement by mathematical induction: $3+7+11+\cdots+(4 n-1)=n(2 n+1)$

Determine the sixth partial sum of the geometric sequence $5, \frac{5}{4}, \frac{5}{16}, \ldots$

Write the statements P₁, P_k , and P_k+1 for the equation $1(1 !)+2(2 !)+\cdots+n(n !)=(n+1) !-1$ as if you were writing a proof by mathematical induction. Do not write a proof.

Find the coefficient of the $x^{3} y^{4}$ term in the expansion of $(x+y) ^{7}$

Find the sum of the infinite series $\sum_{k=1}^{\infty} 4\left(\frac{1}{5}\right)^{k-1}$

The sum of the squares of the first n positive odd integers is $\frac{(2 n-1) (2 n) (2 n+1) }{6}$ Use this expression to find the sum $1^{2}+3^{2}+5^{2}+\cdots+75^{2}$

Sally deposits $\$ 100$ at the end of each month into an account that pays $6 \%$ annual interest compounded monthly. After 5 years of deposits, what is the account balance?

Do the following sequences converge or diverge? If they converge, state the limit. (a) $\left\{\frac{5 n}{2 n+1}\right\}$ (b) $\left\{\frac{n^{3}+1}{2 n+1}\right\}$ (c) $\left\{\frac{3 n^{3}}{4 n^{5}+1}\right\}$

Find the fifteenth term of $(x+2 y)^{20}$ where $x^{20}$ is labelled the zeroth term.

Expand $(5 x-8 y)^{5}$