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
Simplify the polynomial by using foil or box method <br> (2y-1)(y+4)
Sav [38]
So foil
2y² +8y -y-4 = 2y²+7y-4
3 0
3 years ago
Verify the identity (tan x + 1)^2 + (tan x-1)^2= 2 sec^2 x
Elina [12.6K]

(\tan x+1)^2+(\tan x-1)^2=2\sec^2x\\\\\text{use}\ \tan x=\dfrac{\sin x}{\cos x}\\\\L_s=\left(\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\cos x}\right)^2+\left(\dfrac{\sin x}{\cos x}-\dfrac{\cos x}{\cos x}\right)^2\\\\=\left(\dfrac{\sin x+\cos x}{\cos x}\right)^2+\left(\dfrac{\sin x-\cos x}{\cos x}\right)^2\\\\=\dfrac{(\sin x+\cos x)^2}{\cos^2x}+\dfrac{(\sin x-\cos x)^2}{\cos^2x}\\\\\text{use}\ (a\pm b)^2=a^2\pm2ab+b^2

=\dfrac{\sin^2x+2\sin x\cos x+\cos^2}{\cos^2x}+\dfrac{\sin^2x-2\sin x\cos x+\cos^2}{\cos^2x}\\\\=\dfrac{\sin^2x+2\sin x\cos x+\cos^2+\sin^2x-2\sin x\cos x+\cos^2}{\cos^2x}\\\\=\dfrac{2\sin^2x+2\cos^2x}{\cos^2x}=\dfrac{2(\sin^2x+\cos^2x)}{\cos^2x}\\\\\text{use}\ \sin^2x+\cos^2x=1\\\\=\dfrac{2(1)}{\cos^2x}=2\cdot\dfrac{1}{\cos^2x}=2\left(\dfrac{1}{\cos x}\right)^2\\\\\text{use}\ \sec x=\dfrac{1}{\cos x}\\\\=2(\sec^2x)=2\sec^2x=R_s\\\\L_s=R_s\Rightarrow The\ identity

4 0
3 years ago
1. What are the next two terms of the following sequence?
AnnyKZ [126]
All of your marked answers are right
you're very smart (:
6 0
3 years ago
Read 2 more answers
If f(x) =1/4x+3, what is the equation for f^-1 (x)?
ser-zykov [4K]
4x - 12

You can get this by switch the f(x) and the x and then solving for the new f(x) 
7 0
3 years ago
What is the slope of the line? ​
Alik [6]

Answer:

3 or 3/1

Step-by-step explanation:

count the rise over run from exact pints, I used (-1,3) ans (-2,0)

3 0
3 years ago
Read 2 more answers
Other questions:
  • You are making cookies for a bake sale and need to make enough cookies to fill 24 boxes containing 6 cookies each. How many doze
    14·1 answer
  • Find the circumference of a circle with the area of 95.03 m<br><br><img src="https://tex.z-dn.net/?f=c%20%3D%202%5Cpi%20%5C%3A%2
    13·1 answer
  • Simplify 6 - 4[x + 1 - (y+ 3)] +
    13·1 answer
  • A piece of cheese is in the shape of an isosceles right triangular prism. The triangular base is 4 cm and the height is 2 cm. Th
    9·1 answer
  • Write 4 and 1/4 as a percent
    8·1 answer
  • In minutes, how much is 2 hours?
    5·2 answers
  • PLEASE HELP!!!!!!!!!!!!
    5·1 answer
  • 2y +9 =x 9x- 8y=31<br> System of substitution
    10·2 answers
  • Hm what is it?????? Answer pls
    11·1 answer
  • Find the greatest common factor and the least common multiple of 16 and 20. The prime factorizations of each number are given.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!