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
lions [1.4K]
3 years ago
15

Using the Principle of Mathematical Induction, prove that . n ^2 − n is even for n ≥ 1​

Mathematics
1 answer:
Tamiku [17]3 years ago
8 0

Base case: if <em>n</em> = 1, then

1² - 1 = 0

which is even.

Induction hypothesis: assume the statement is true for <em>n</em> = <em>k</em>, namely that <em>k</em> ² - <em>k</em> is even. This means that <em>k</em> ² - <em>k</em> = 2<em>m</em> for some integer <em>m</em>.

Induction step: show that the assumption implies (<em>k</em> + 1)² - (<em>k</em> + 1) is also even. We have

(<em>k</em> + 1)² - (<em>k</em> + 1) = <em>k</em> ² + 2<em>k</em> + 1 - <em>k</em> - 1

… = (<em>k</em> ² - <em>k</em>) + 2<em>k</em>

… = 2<em>m</em> + 2<em>k</em>

… = 2 (<em>m</em> + <em>k</em>)

which is clearly even. QED

You might be interested in
How to write an equation in standard form for a circle of other radius 3 centered at the origin
Y_Kistochka [10]

Answer:

What is the diameter of a circle if the radius is 3

3 0
2 years ago
Read 2 more answers
5.
Sphinxa [80]

Answer:

1961

Step-by-step explanation:

circumference = 2\pi R\\157 = 2\pi R\\157 = 2\pi* R\\R = 157 / 2\pi \\\\area = \pi R^{2} \\area = \pi* (157 / 2\pi)^{2} \\area = \frac{24649}{4\pi} \\area = 1961 \\

ok, I did smth wrong

I hope you can find error

5 0
3 years ago
The f(x)=2^x and g(x)=f(x+k). If k=-5, what can be concluded about the graph of g(x).
statuscvo [17]

Answer:

A and C

Step-by-step explanation:

so the final g(x) would be y=2^x+5

A because of the +5

C because there are no parenthesis and therefor cant be a horizontal shift

7 0
3 years ago
Read 2 more answers
Roger owns a one-acre piece of land. The length of the land is 484 feet. What is the width of his property?
shepuryov [24]

Answer:

W = 90 ft

Step-by-step explanation:

1 acre = 43,560 ft^2

A (area) = L (length) x W (width),

L = 484

A = 43560

W = A/L

= 43560/484

= 90 ft

8 0
3 years ago
Help and if you actually answer I'll give you brainliest​
schepotkina [342]

Answer:

1) 2/3

2) 1/2

3) 4/5

4) 3/4

Step-by-step explanation:

4/6 can be divided by 2 to get 2/3 in its simplest

2/4 can be divided by 2 to get 1/2 in its simplest

12/15 can be divided by 3 to get 4/5 in its simplest

6/8 can be divided by 2 to get 3/4 in its simplest

6 0
2 years ago
Read 2 more answers
Other questions:
  • Lenny loanz borrowed $8000 for a used car and paid 9
    7·1 answer
  • Marna is playing a game where you score -5 points each time you guess the correct answer. The goal is to get the lowest score. T
    14·2 answers
  • GEOMETRY. PLEASE INCLUDE CLEAR ANSWER AND EXPLANATION
    11·1 answer
  • What would happen if grass and plants disappeared from the food chain?
    10·2 answers
  • (3x + 7)5<br> Simply the expression
    7·2 answers
  • Math help pleaseeeeeeeeeeeeeeeeeeeeeeee
    11·1 answer
  • Robert wants to average a 92% in math class his first three marking periods scores are 89%,95%, and 90% what does he need to get
    5·2 answers
  • Calculate the measure of ∠ PMU and ∠ UPM.<br><br> m∠ PMU =<br> m∠ UPM =
    5·2 answers
  • A package of 6 pairs of insulated gloves costs $35.94. What is the unit price of the pairs of gloves?
    11·2 answers
  • . If y varies directly with x when y = -4 and x = 10, determine y when x = 2. Write and equation for the direct variation and de
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!