1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Rudik [331]
3 years ago
6

Problems 2.21, 2.22, 2.23

Mathematics
1 answer:
RoseWind [281]3 years ago
7 0

Statements can be proved by contrapositive, contradiction or by induction.

  • <em>2.21 and 2.23 are proved by contrapositive</em>
  • <em>2.22 is proved by induction</em>

<u />

<u />

<u>2.21: If </u>n^3<u> is even, then n is even (By contrapositive)</u>

The contrapositive of the above statement is that:

<em>If n is odd, then  </em>n^3<em> is odd</em>

Represent the value of n as:

n = 2k + 1, where k \ge 0

Take the cube of both sides

n^3 = (2k + 1)^3

Expand

n^3 = 8k^3 + 6k^2 + 6k + 1

Group

n^3 =[ 8k^3 + 6k^2 + 6k] + 1

Factor out 2

n^3 =2[4k^3 + 3k^2 + 3k] + 1

Assume w is an integer; where:

w =4k^3 + 3k^2 + 3k

So, we have:

n^3 =2w + 1

The constant term (i.e. 1) means that n^3 is odd.

Hence, the statement has been proved by contrapositive.

<em>i.e. If n is odd, then  </em>n^3<em> is odd</em>

<u />

<u>2.22  </u>3n + 4<u> is even, if and only if n is even</u>

We have: 3n + 4<u />

<u />

Assume that: n = 2k + 2 for k \ge 0

So, we have:

3n + 4 = 3(2k + 2) + 4

Open bracket

3n + 4 = 6k + 6 + 4

3n + 4 = 6k + 10

Factorize

3n + 4 = 2(3k + 5)

The factor of 2 means that 3n + 4 is even.

<em>Hence, </em>3n + 4<em> is even, if and only if n is even </em>

<em />

<u />

<u>2.22: </u>s \ne -1<u> and </u>t \ne -1<u>, then </u>s + t + st \ne -1<u />

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If s = -1 and t=-1, then s + t + st = -1

We have:

s + t + st = -1

Substitute the values of s and t in: s + t + st = -1

-1 -1 -1 \times -1 = -1

-1 -1 + 1 = -1

-1 = -1

Hence, by contrapositive:

If s = -1 and t=-1, then s + t + st = -1

Read more about proofs  at:

brainly.com/question/19643658

You might be interested in
ASAP<br><br> - How did you get the answer?
Tatiana [17]

Answer:

quizlet .

Step-by-step explanation:

7 0
2 years ago
Read 2 more answers
Find the derivative of the inverse function of the original function f(x)=(x+1)^3-5
____ [38]

Answer:

Step-by-step explanation:

let f(x)=y

y=(x+1)^{3} -5\\flip ~x~and~y\\x=(y+1)^{3}-5\\(y+1)^{3}=x+5\\y=(x+5)^{\frac{1}{3} } -1\\or~f^{-1}(x)=(x+5)^{\frac{1}{3} } -1\\(f^{-1})'(x)=\frac{1}{3} (x+5)^{\frac{-2}{3} }

4 0
3 years ago
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!
Leno4ka [110]

Add a point at (0.-2) then go down 3 and put another point connect the dots thats it

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
PLZ HELP!!!!! WILL MARK!!!
Alex777 [14]
4x^2 +15x + 13 tyats would be tour answrr
6 0
2 years ago
4) A bank is offering 3.5% simple interest on a savings account. If you deposit
Gnom [1K]

Answer:

B. $840

Step-by-step explanation:

Given that

Rate of simple interest = 3.5%

Principal amount = $12,000

Time = 2 year

To find:

Interest earned = ?

Solution:

Formula for Simple Interest is given as:

S.I. = \dfrac{PRT}{100}

Where P is the principal amount

R is the annual Rate of interest

T is the time for which principal was invested.

Here, we are given that:

P = $12,000

R = 3.5%

T = 2 years

Putting all the values in the formula:

SI= \dfrac{12000\times 3.5 \times 2}{100}\\\Rightarrow SI= 12\times 35 \times 2\\\Rightarrow SI  = \$840

So, answer is:

B. $840

4 0
3 years ago
Other questions:
  • What are the answer to these questions?
    12·1 answer
  • Y" - 4y = ((x^2) - 3) sin 2x
    6·1 answer
  • 22 less than the quotient of an unknown number and 35 is -25
    14·1 answer
  • Please Help!!! Emergency!!! 50 points!!!
    7·2 answers
  • A.<br> B.<br> C.<br> D.<br> ????????????
    15·1 answer
  • Which of the following mapping diagrams does not represent a function?
    14·1 answer
  • 5(2)^x=40 what is the value of x
    14·2 answers
  • A motorcycle raced around a track at a speed of 110 miles per hour.
    12·1 answer
  • I need help:) kinda urgent but ILL MARK brainliest!! :)
    15·1 answer
  • -7/10 + 7 2/3<br> show all ur work
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!